II 


ELEMENTS 


INTEGRAL    CALCULUS; 


KEY  TO    THE  SOLUTION  OF  DIFFERENTIAL 

EQUATIONS,    AND   A   SHORT   TABLE 

OF  INTEGRALS. 


WILLIAM   ELWOOD   BYERLV,   Ph.D., 

PROFESSOR   OF    MATHEMATICS    IN    HARVARD    UNIVERSITY. 


SEdcrXD  EDITION,   RKVI^EV  AXD  EX  LA  IKIED. 


GTNN  &  COMPANY 

BOSTON  •  NEW  YORK  •   CHICAGO  •  LONDON 


//.,       y     ■     :^    .        -^  .  ...     ,     ^  I 


-t 


-^i^^^-exc^^^ 


Entered  according  to  Act  of  Congress,  in  the  year  1888,  by 

WILLIAM    KLWOOn   B-i-EKLY, 

in  the  OUlcc  of  the  Librarian  of  Congress  at  Washington. 


'0  //l-f-t^ 


All  RiaiiTS  Reservkd. 


/  /  r,lNN    .V    COMI'ANI 


PklHTUKS  •  BOSTON  •  U.S.A. 


y^ 


^^oQfe.  /y/^//A^>7^/3yr^^^^^'^^i/ 


PREFACE, 


The  following  volume  is  a  sequel  to  m)-  treatise  on  the 
Differential  Calculus,  aud,  like  that,  is  written  as  a  text-hook. 
The  last  chapter,  however,  a  Key  to  the  Solution  of  Ditlercntial 
Equations,  ma^'  prove  of  service  to  working  mathematicians. 

I  have  used  freeh'  the  worlcs  of  Bertrand,  Beujamin  Peirce, 
Todhunter,  and  Boole  ;  aud  1  am  much  indebted  to  Professor 
J.  M.  Peirce  for  criticisms  and  suggestions. 

I  refer  constantly  to  ni}-  work  on  the  Differential  Calculus 
as  Volume  I. ;  and  for  the  sake  of  convenience  I  have  added 
Chapter  V.  of  that  l)ook,  which  treats  of  Integration,  as  an 
appendix  to  the  present  volume. 

W.  E.   BYEliLY. 

C'AMUiaUCJE,  1881. 


54243S 


PREFACE   TO   SECOND   EDITION. 


In  enlarging  my  Integral  Calculus  I  have  used  freely 
Scljlomilch's  '•  Compendium  der  IlOliereu  Analysis,"  Cayley's 
"  Elliptic  Functions,"  Meyer's  "  Hestininite  Integrale,"  For- 
syth's "  Ditferential  Equations,"  and  Williamson's  "Integral 
Calculus." 

The  chapter  on  Theory  of  Functions  was  sketched  out  and 
in  part  written  by  Professor  B.  O.  Peirce,  to  whom  I  am 
greatly  indebted  for  numerous  valuable  suggestions  touching 
other  portions  of  the  book,  and  who  has  kindly  allowed  me 
to  have  his  Short  Table  of  Integrals  bound  in  with  this  volume. 

\V.  K.  BYEULY 
Cambkiuue,  1888. 


ANALYTICAL   TABLE   OF   COiXTENTS. 


CHAPTER 


SYMBOLS   OF   OPEIJATION. 
Article.  Page 

1.  Functional  symbols  rci;;irck'(!  ;is  syjH/jo/.'?  o/op^m<jo?i 1 

2.  Compound  I'liuctiou;  compouud  operation i 

3.  Commutative  or  relatively  free  operations 1 

4.  Distributive  or  linear  operations 2 

5.  The  compounds  of  dismftjtiiye  operations  lire  distributive  .    .    .  2 

6.  Symbolic  exponents 2 

7.  The  law  of  indices 2 

8.  The  interpretation  of  a  zero  exponent 3 

9.  The  interpretation  of  a  negative  exponent 3 

10.    When  operations  are  commutative  and  distributive,  the  sym- 
bols which  represent  them  may  be  combined  as  if  they  were 

algebraic  quantities 3 


CHAPTER    II. 

IMAGIXAUIKS. 

11.  Usual  definition  of  an  imftf/inanj.    Imaginaries  first  forced  upon 

our  attention  in  connection  with  quadratic  equations  ....  5 

12.  Treatment  of  imaginaries  purely  arbitrary  and  conventional  .    .  (5 

13.  V—1  defined  as  a  symbol  of  operation fi 

14.  The  rules  in  accordance  with  Avhich  the  symbol  V—1  is  used. 

V^  (listrihutivp  and  commntativr  with  symbols  of  quantity  .      7 

15.  Interpretation  of  powers  of  v'— 1 7 

IG.    Imaginary  roots  of  a  quadratic <^ 

17.  Typical  form  of  an  imaginary.     E<iual  iniairinarics 8 

18.  Geometrical  representation  of  an  imau:inary.     7?ra/.s  and  purr 

imaf/iufiries.     An  interpretation  of  the  operation  \/—  I    .    .    .      a 

19.  The  sum.  the  product,  and   the  quotient  of  two  Imaginaries, 

a-\,bV^l  and  c  +  dV^,  arc  imaginaries  of  the  typical 
form 10 


VI  INTEGRAL   CALCULUS. 

Artidc  race 

20.  Second  typical  form  r(cos<p+  \/—ls\n(p).     Modulus  and  rtr(fu- 

mi'ut.     Absolute  vulur  of  an  iiiiajj;inary.     Examples 10 

21.  The  modiihis  of  the  sum  of  two  iniagiuaries  is  never  j^ieater 

than  the  sum  of  tlu'ir  moduli 11 

22.  .Modulus  and  argument  of  the  product  of  imagiuarie.s 12 

23.  Modulus  and  argument  of  the  quotient  of  two  iniagiuaries     .    .  13 

24.  Modulus  and  argument  of  a  power  of  an  imigiuary 13 

25.  Modulus  and  argument  of  a  root  of  an  imaginary.     Example    .  U 

26.  Relation  between  the  n  nth  roots  of  a  real  or  au  imaginary    .    .  14 

27.  The  imaginary  roots  of  1  and  —  1.     Examples 15 

28.  (Ju)iju(jat^  imaginarics.     Examples 17 

29.  Transcendental  functions  of  au  imagiuary  variable  best  defined 

by  the  aid  of  series 17 

30.  Convergency  of  a  series  containing  imaginary  terms 18 

31.  Exponential  functions  of  an  imaginary.     Definition  of  c*  where 

z  is  imaginary 19 

32.  The  laio  of  indices  holds  for  imaginary  exponentials.    Example  20 

33.  Lofjarithmic  functions  of  an  imaginary.      Definition  of  logs. 

Log  2  a  ;)eno(Z<c  finictiou.     Example 21 

34.  Triyonometric  functions  of  au  imagiuary.     Definition  of  sius 

and  cos  z.     Example 22 

35.  Sin  z  and  cos  z  expressed  in  exponential  form.     The  fundamen- 

tal fornuilas  of  Trigonometry  hold  for  iraaginaries  as  well  as 

for  reals.     Examples 22 

3(;.  Differentiation  of  Functions  of  Imaginary  Variables.  The  de- 
rivative of  a  function  of  au  imaginary  is  in  general  indetermi- 
nate    24 

37.  In  differentiating,  we  may  treat  the  V^  like  a  constant  factor. 
Example.     Two  forms  of  the  differential  of  the  independent 

variable 24 

3.S.    Differentiation  of  a  power  of ," 25 

39.  Differentiation  of  e*.     Example 20 

40.  Differentiation  of  log.? 2(5 

41.  Differentiation  of  sin. i- and  cos z 2() 

42.  Formulas  for  direct  inteijratioa   (I.,  .\rl.   74)    iioUl  when  ./•   is 

imaginary 27 

l.'i.    Ilyjicrliolic  Functions 27 

It.    Examples.     Properties  of  Hyperbolic  Functions 28 

I.").    Differentiation  of  Hyperbolic  Functions 2S 

HI.    Anti-hyperbolic  functions.     Examples 28 

47.    Anti-hyi)erbolic  flnictlons  expressed  as  logarithms 29 

18.    Formulas  for  the  direct  integration  of  some  irratioMa!    forms. 

Examples 30 


TABLE  OF  CONTENTS. 


CHAPTEU  III. 

GEXr.lJAL    MKTIIOU.S    OK    IX TKC.K.V TING. 
Article.  Page. 

49.  Integral  regarded  as  the  inverse  of  a  differential 32 

50.  Ufx  is  any  function  whatever  of  x,fx.dx  has  an  integral,  and 

but  one,  except  for  the  presence  of  an  arbitrary  constant  .    .    32 
61.   A  definite  integral  contains  no  arbitrary  constant,  and  is  a  func- 
tion of  tlie  values  between  which  the  sum  is  taken.     Exam- 
ples   aa 

52.  Definite  integral  of  a  f?<s('o«?i«?(oif»"  function 33 

53.  Formulas  for  direct  integration 34 

54.  Integration  by  substitution.     Examples 3G 

55.  Integration  by  j)arts.     Examples.     Miscellaneous  examples  in 

integration 37 


CII.VPTER    IV. 

RATION  A  r,    FRACTIONS. 

66.   Integration  of  a  rational  algrbraic  pohinnmial.     Kational  frac- 
tions, proper  and  improper 40 

57.  Every  proper  rational  fraction  can  be  reduced  to  a  sum  of  .sim- 

pler fractions  with  constant  numerators 40 

58.  Determination  of  the  numerators  of  the  partial  fractions  by 

indirect  methods.     Examples V2 

59.  Direct  determination  of  the  numerators  of  the  partial  fractions  43 

60.  Illustrative  examples 45 

61.  Case  where  some  of  the  roots  uf  tht-  denominator  are  imaginary   .  4'» 

02.    Illustrative  example 47 

63.    Integration  of  the  partial  fractions.     Exanii)les 4'.» 


CMAl'TEll    V. 

IJKDUCTION    FORMULAS. 

64.  Formulas  for  raising  or  lowering  the  exponents  in  tin-  form 

.r'"-'(rt  + ''.r»)''fZ.t 

65.  Consideration  of  special  cases.     Examph's 


01 


INTEGKAL   CALCULUS. 


CHAriKK    VL 

lUICATIitNAI.    KOHM8. 
Altieto.  rage 

fifi.    Intcgrntion  of  the  fonii  /(.r,  \^a  +  hx)rlr.     Examples 5(J 

CT.  Iiit.';:nitioii  of  tlie  form  /(z,  \^c-\-  \/n  +  hx)dx.     Examples    .    .  57 

r.«.   Inli-;;riitioii  of  llie  form  /(r,  y/a  +  bx  +  cx*)(Lr. o7 

G9.   IlluHtrative  example.     Examples '/J 

71).    Integration  of  the  form  rf  3^, -vf^^^^lrfj.    Example    .... 

•    V         >  /X  +  HI  / 

71.  .Application  of  the  liidxution  Formulas  of  Chapter  V.  to  irra- 

tional forms.     Examples CI 

72.  A    function   rendered    irrational    tliroufili    the   presence   under 

the  ratlical  si;:n  of  a  polynomial  of  higher  decree  than  the 
second  cauuut  ordinarily  he  integrated.     Elliptic  Integrals    .    Ii2 


CHAPTER    VII. 

TRANSCKXliKXTAL   Kl'.NCTIOXS. 

73.  Use  of  the  method  of  InUgmtion  by  Parts.     Examples     .    .    .    .  G3 

74.  Hediiction  F(»rmulas  for  .siu''a:  and  cos-'j-.     Examples (54 

75.  Integration  of  (sin-' J-)" (?r.     Examples (15 

76.  r.se  of  the  methoil  of  Integration  by  Substitution CO 

77.  Substitution  of  r=tan^  In  Integrating  trigonometric  forms     .  67 

78.  Integrathm  of  sin"/cos''/-.(?j-.     E.vamples C8 

79.  Kcduction  formulas  for  ftan-x.dj;  and  i-—^-     Examples.   .  69 

J  Jtan'-z 


CHATTER   VIII. 

r»KKINITK    INTK.(iKAI.S. 

80.    Dednltlon  of  a  (hjlniu-   intnjral   as    the   Hndt  of  a  sum   of 

Inllnlteslnmls 7] 

M.  Conij)ntallon  of  a  tieflnlte  integral  as  the  limit  (.f  a  sum.  Illus- 
trative exiin>|)les.     Examples 72 

82.    Usual  nieth.xl  of    obtaining  the  value  of  a  dellnite  integnil. 

Cautlou  couceruiiig  multiple-valued  luucUous.    Examples  .     76 


TABLE   OF   CONTENTS. 


IX 


Article  ,^  Tuk-f. 

83.  Consideration  of  tlio  nature  of  the  value  of  |  fx  .  dx  wlien  fx 

becomes  infinite  for  a;  =  a,  or  x  =  6,  or  fur  .some  value  of  x 
between  a  and  6.     Illustrative  examples 80 

84.  Maximum-Minimum  Theorem.     Test  that  must  be  satisfied  in 


order  that  \  fx  .dx  may  be  finite  and  determinate  if  fx  is 

infinite  for  some  value  of  x  between  a  and  b.     Illustrative 
examples.     Examples 82 


fx  .  dx  shall  be  finite 


85.  Meaning  of  |  fx  .  dx.     Condition  that  | 

and  determinate 88 

86.  Proof  that  certain  important  definite  integrals  of  the  form 

fx  .  dx  are  finite  and  determinate.     Examples     ....     89 


I 


obtained 


87.  Application    of    reduction    formulas    to    definite    integrals. 

Examples 01 

88.  Application  of  the  method  of  integration  by  sribstitution  to 

definite  integrals.     Illustrative  examples.     Example  .    .     .     O'j 

89.  Differentiation  of  a  definite  integral.     Examples 9G 

90.  Many  ingenious  methods  of  finding  the  values  of  definite  inte- 

grals are  valid  only  in  case  the  integral  is  finite  and  de- 
terminate     100 

91.  Integration  by  development  in  series.     Examples 100 

92.  Values  of  |    log  sina;.(?x,  |     e  *V?x,  I     " '-dx 

c/o                   •      «/o                »/o         ^ 
by  ingenious  devices.     Examples 102 

93.  Differentiation  or  integration  with  respect  to  a  quantity  wliith 

is  independent  of  x.     Examples 10.') 

94.  Additional  illustrative  examples.     Examples 10(1 

95.  Introduction  of  imaginary  constants HW 

96    The  Gamma  Function 109 

97.  Table  giving  logr(H)  from  n=  1  to  n  =  2.     Definite  integrals 

expressed  as  Gamma  Functions Ill 

98.  The  Beta  Function.     Formula  connecting  the  Beta  Function 

with  the  Ganinia  Function.     Value  of  r(.]) 113 

99.  More    definite    integrals    expressed    as    Gamma  Functions. 

Examples Hi 


INTEGKAL   CALCULUS. 


rHAl'TF.U    IX. 
i.k.V(;tiis  or  iikves. 


Article.  P«ge- 

KM).  Formulas  for  slnr  ami  cos  r  In  terin.s  of  the  lenjrth  of  tin  arc  117 

101.  The  e(|uation  of  the  Crt/<'Hrtry  obtaiiu'd.     E.vaniple     .     .     .     .117 

102.  The  cijnntion  of  the  7'r</r/nV.     E.xamples 11!) 

103.  Leiifrth  of  an  arc  in  rectiiiiLrulnr  coordinates 121 

104.  Lenfilh  t)f  the  arc  of  the  C>7ry/(/.     E.xample 122 

105.  Another  method  of  rectifying  the  Cycloid 123 

IOC.  Ucctillcation  of  the  Epiryrhiid.     Examples 12"5 

107.  Arc  of  the  A7/i;w.     Auxiliary  an^le.     Example 124 

108.  Lenfrth  of  an  arc.     I'olar  coordinates 125 

109.  Equation  of  the  Logarithmic.  Spiral 125 

110.  Kectillcation  of  the  Lofrarithmic  Spiral.     Examples   .     .     .     .  120 

111.  Rectification  of  the  tVin/joiWe 12() 

112.  Itirolutes.     Illustrative  example.     Example 127 

ll.'l.  The  i/irfi/wr^  of  the  Cycloid.     Example 129 

114.  Fiitrinsic  equation  of  a  curve.     Example 130 

115.  Intrinsic  equation  of  the  Epicycloid.     Example i;U 

Ufi.  Intrinsic  equation  of  tlie  Lofraritlnnic  Spiral 182 

117.  Method  of  obtaininj?  tlie  intrinsic  equation  from  tlie  equation 

in  rectanifular  coordinates.     Examples 1.32 

118.  Intrinsic  e<|uation  of  aiwi«/M<fi 134 

lit).  Illustrative  exam|»les.     Examples 134 

120.  The  e volute  of  an  JJpicijclnid.     Example 135 

121.  The  intrinsic  e(|uation  of  an  involute.     Illustrative  examples    136 

122.  Limiting  form  api)roached  by  an  involute  of  an  involute    .     .  137 

123.  Metliod  of  obtainin<;  the  equation  in  rectangular  coordinates 

from  the  intrinsic  eipiation.     Illustrative  example  ....   138 

124.  Kectillcatiou  of  C'Krccs  i«  (Sy"i<( .     Examples 130 


CIIAI'IKK    X. 


125.  A II as  expressed  as  dclluih'  inicirnils,  rectangular  coc'inlinatcs. 

Examples lU 

126.  ArcnH  expressed  as  delinite  integrals,  polar  counlinatcs  .  143 

127.  Area  between  the  catenary  and  the  axis 143 

12«.    Area  l>i-t\veen  tlic  tructrix  and  the  axis.     Example      ....  143 


TABLE   OK   CONTKXTS.  Xl 

Article.  Page. 

12',).    Area  between  a  curve  and  its  asymptote.     Examples      .     .     .144 

130.  Area  of  circle  obtained  by  the  aid  of  an  au.xiliary  anjjle.     Ex- 

amples   145 

131.  Area  between  two  curves  (rect.  coor.).     Examples    ....  146 

132.  Areas  in  Polar  Coordinates.     Examples 147 

133.  Problems  in  areas  can  often  be  simplified  by  transformation 

of  coordinates.     Examples \',0 

134.  Area  between  a  curve  and  its  evolute.     Examples loO 

135.  Holditch's  Theorem.     Examples 151 

ISG.   Areas  by  a  double  intrr/ratiou  (rect.  coor.) 153 

137.  Illustrative  examples.     Examples. l.')4 

138.  Areaahy  a  double  intei/ration  (polar  coor.).    Example    .    .    .155 

CHAPTER   XI. 

ARE.VS   OK   SUHFACES. 

139.  Area  o{  a  surface  of  1-evoIution  (rect.  coor.).    Example  .     .    .  157 

140.  Illustrative  examples.     Examples 158 

141.  Area  of  a  surface  of  revolution  by  transformation  of  coordi- 

nates.    Example 159 

142.  Area  of  a  s?(r/«ce  o/rewoZ?/<?'>??.  (polar  coor.).     Examples    .     .  161 

143.  Area  of  a  cylindncal  siirfacp.     Examples 161 

144.  Area  of  any  surface  by  a  double  integration 104 

14.").    Illustrative  example.     Examples 107 

\\C^.    Illustrative  example  requiriug  transformation  to  i)olar  coordi- 
nates.   Examples 109 

CHAPTER   XII. 

VOI.r.MKS. 

147.   Volume  by  a  single  integration.     Example 172 

US.    Volume  of  a  co«oiV/.     Examples 173 

14'J.    Volume  of  an  ellipsoid.     Examples 174 

150.  \o\\xmeoi  a  solid  of  revolution.     Single  integration.     Exam- 

ples     175 

151.  \o\\\mc  oi  a  solid  of  revohition.    Double  integration.     Exam- 

ples  176 

152.  \o\\\me  of  a  solid  of  revolution.     Polar  formula.     Example     .   178 

153.  Volume  of  any  solid.    Triple  integration.     Rectangular  coor- 

dinates.   Examples 179 

154.  Volume  of  any  solid.    Triple  integration.     Polar  ccxirdinales. 

Examples 183 


INTEGIiAL   CALCULUS. 


CHAriKK    XIII. 


CKNTKKS    UK    tiUAVITY. 
Article.  Page. 

155.   Centre  of  Gravity  (U'llned .184 

150.   General  lonmiliL"^  for  the  eoorilinates  of  the  Centre  of  Gravity 

of  any  mass.     Example 184 

157.  Centre  of  Gravity  of  a  homogeneous  body 186 

158.  Centre  of  Gravity  of  a  ;)/rtHe  arprt.     Example.s 186 

159.  Centre  of  Gravity  of  a /jomf)(;rc>ietiHs  »o/i(i  of  revolution.     Ex- 

amples   189 

160.  Centre  of  Gravity  of  an  arc ;  of  a  surface  of  revolution.     Ex- 

amples            .  191 

161.  Properties  of  Guldin.     Examples 192 


ClIAPTEU   XIV. 

LINE,   Sl-UKACK,    AND   Sl'ACK   INTEORALS. 

IC,2.    Point  function.     CV>h</hh(7(/ of  a  point  function 194 

IG.'J.    Linc-intf'tjnil,  surficc-intcyrdl,  and  spdcp-intcyrctl  of   a   point 

function 194 

164.  Value  of  a  line,  surface,  or  space  integral  independent  of  the 
position  in  each  element  of  the  point  at  which  the  value  of 
the  function  is  taken 195 

16r>.  Value  of  a  line,  surface,  or  space  integral  independent  of  the 
manner  in  which  the  line,  surface,  or  space  is  l)rok('n  up 
Into  inllnitesimal  elements 195 

166.   Geometrical  representation  of  a  line-integral  along  a  plane 

curve,  and  a  surface-integral  over  a  i)laue  surface  .     .         .197 

367.   Moments  of  inertia.     Examples ' 197 

168.  Uelatlon  between  a  surface-integral  over  a  plane  surface,  and 

a  line-integral  along  the  curve  bounding  the  surface.     Ex- 
ample;      'jcO 

169.  Illustrative  example.     Examples 202 

170.  Another  form  of  the  relation  established  in  .\rt.  168.    Example  203 

171.  Relation  between  a  si)ace-lntegral  taken  throughout  a  given 

space  and  a  surface-Integral  over  the  surface  bounding  the 
spa<-e.     Example 203 

172.  Illustrative  example.    Example     .    .         205 


T^UiLl-:   OF   CONTENTS.  XXii 


CHAPTER  XV. 

MEAN   VALUE   AND    I'KOBABILITY. 
Article.  Page. 

173.  References 206 

174.  Mean  value  of  a  continnonsly  varying  quantity.    The  mean  dis- 

tance of  all  the  points  of  the  circumferences  of  a  circle  from 
a  fixed  point  on  the  circumference.  The  mean  di.-<tanri'  of 
points  on  the  surface  of  a  circle  from  a  tixetl  point  on  the 
circumference.  The  mean  distance  of  points  on  the  surface 
of  a  square  from  a  corner  of  the  square.  The  mean  distance 
between  two  points  within  a  given  circle 200 

175.  Problems  in  the  application  of  the  Integral  Calculus  to  proba- 

bilities.   Random  straight  lines.     Examples 208 


CHAPTER  XVI. 

ELLIPTIC   IXTPXiUALS. 

176.  Motion  of  a  simple  pendulum.    Vibration.    Complete  revolution  2\d 

177.  The  length  of  an  arc  of  an  Ellipse 217 

178.  Algebraic  forms  of  the  Elliptic  Integrals  of  the  llrst,  second, 

and  third  class.     Modulus.     Parameter 217 

179.  Trigonometric  forms  of  the  Elliptic  Integrals.     Amplitttde. 

Delta.     Complementary  Modulus 218 

180.  Landen's  Transformation.    Reduction  formula  bj'  which  we 

can  increase  the  modulus  and  diminish  the  amplitude  of  an 
Elliptic  Integral  of  the  lirst  class.  A  method  of  computing 
F{k,<p) 219 

181.  Reduction  formula  for  diminishing  the  modulus  and  increas- 

ing the  amplitude  of  an  Elliptic  Integral  of  the  first  class. 
A  second  method  of  computing  F{k,  <f>) 222 

182.  Actual  computation  of  f(—,   -\  and  f(—.  -")....  223 

183.  Landen's  Transformation.    Reduction  formula  by  which  we 

can  increase  the  modulus  and  diminish  the  amplitude  of  an 
Elliptic  Integral  of  the  second  class.  A  method  of  com- 
puting E{k,p) 220 

184.  A  reduction  formula  for  diminishing  the  modulus   and    in- 

creasing tlie  amplitude  of  an  Elliptic  Integral  of  the  second 
class.    A  secoud  method  of  computing  E  {k,  <p) 229 


XIV  INTEGRAL   CALCULUS. 

Article.  _  P«ge 

V2      ,r\    „„^     ^/V2 


185. 


Actual  compuliitic.n  of /•;['^^,    -\  and  Z:/'^,   ']   .    .     .     .231 

18r>.  An  Elliptic  Integral  of  the  llrst  or  second  class,  whose  anipli- 
tiule  is  greater  than  -,  can  be  made  to  depend  upon  one 
whose  amplitude  is  less  than  ^,  and  upon  tliu  correspond- 
ing complete  Elliptic  Integral 235 

187.   Three-place  table  of  Elliptic  Integrals  of  the  llrst  class  and 

the  secoiul  class 237 

18S.  Addition  Formulas.  Functions  defined  by  the  aid  of  definite 
integrals,    logs-,  sin  'x,  tan'' r,  F(k,x),  E  (k,:r).    Addition 

formula  for  log  x 239 

l.s'.i.    Addition  formulas  for  sin'. r  and  tiin' .r 241 

i;iU.    Addition  formula  for  J^(t,a;) .     .     .  243 

I'Jl.    .Vnalogy  between  log^M,  sin?/,  tan  7(.  and /^'^  (i*,  ?<)  .     .    .     .245 
r.»2.   The  Kllijidr  Fttnrtions,  sum,  cn?(,  and  dn ?/.    Their  analogy 
with  Trigonometric  Functions.     Formulas  connecting  the 

Elliptic  Functions  of  a  single  (juantity 246 

Ii».'5.    Formulas  for  Elliptic  Functions  of  (u-j-f)  and  (u  —  v)    .     .     .  248 
WW.   Formulas  for  Elliptic  Functions  of  2  w 250 

195.    Formulas  for  Elliptic  Functions  of  " 250 

19G.    /V')-i'o(/iV{7»/ of  the  Elliptic  Functions.     Real  period,  4  A'.     .     ,251 

197.  Elliptic  Functioub  of  a  pure  imaginary.    Jacobi's  Transforma- 

tion.   Elliptic  Functions  have  an  imaginary  period,  4  K'V—l. 
Table  of  values  of  Elliptic  Functions  having  the  modulus 

V2 253 

2 

198.  The  Elliptic  Integral  of  the  second  class  expressed  in  terms 

of  Elliptic  Functions.     Addition  formula  for  Elliptic  Inte- 
grals of  the  second  class 250 

199.  Application  to  the  rectification  of  "the  f.fmniscatc     Examples. 

Bisection  of  the  arc  t)f  a  t|ua(lrant  of  the  Lemniscate  .    .    .  258 

200.  Kectillcation  f)f  the  Ellipse.     Examples 261 

201.  Use  of  the  Addition  Formula  in  dealing  with  Elliptic  arcs. 

FngnanVa  Point.     Examples 262 

202.  Rectification  of  the  Jli/prrhula.     Examples 264 

203.  The  simple  pendulum.    Examples 267 

204.  Direct  integration  of  irrational  Algebraic  functioii.s.    Examples  275 

205.  Transformation  of  Elliptic  Integrals.     Examples 281 

206.  Integration  of  Elliptic  Functions.     Examples 282 


TABLE   OF   CONTENTS.  XV 


CHAPTER   XVII. 

INTRODUCTIOX   TO   THE   TIIP:OKY    OK    KIXCTIOXS. 
Article.  Page. 

207.  Single-vahied  functions.     Multiple-valued  functions    ....  283 

208.  Importance  of  the  graphical  representation  of  iniagliiaries. 

Complex  quantity 283 

209.  When  a  complex  variable  is  said  to  varj-  continuouslj'^    .    .    .  284 

210.  A  continuuHs  function  of  a  complex  variable.     Critical  values   284 

211.  Criterion  that  a  function  shall  have  a  determinate  derivative. 

Monoyenic  functions 285 

212.  Any  function  involving  z  as  a  whole  is  a  monogenic  function    288 

213.  Conjugate  functions.      Their  use  as  solutions  of   Laplace's 

Equation.     Example 288 

214.  Preset-cation  of  angles 290 

215.  If  two  paths  traced  by  the  point  representing  the  variable 

have  a  common  beginning  and  a  common  end,  and  do  not 
enclose  a  critical  point,  tlie  corresponding  paths  traced  by 
the  point  representing  the  function  and  having  a  common 
beginning  will  have  a  common  end 292 

216.  Examples  where  the. paths  traced  by  the  point  representing 

the  variable  enclose  a  critical  point 294 

217.  Critical  points  at  which  the  derivative  of  the  function  is  zero 

or  infinite  are  to  be  avoided.  Branchpoints.  Uolomorphic 
functions 295 

218.  Definite  integral  of  a  function  of  a  complex  variable  defined. 

Such  an  integral  is  generally  indeterminate,  and  depends 
upon  the  patli  by  which  the  point  representing  the  variable 
passes  from  the  lower  limit  to  the  upper  limit  of  the  integral  297 

219.  If  the  function  is  holomorphic,  tlie  definite  integral  is  in  gen- 

eral determinate 299 

220.  Tile  integral  around  a  closed  contour,  embracing  a  point  at 

Avhich  the  function  is  infinite 300 

221.  Illustrative  examples 301 

222.  Convcrgency  of  the  series  o])tained  by  integrating  tlie  terms 

of  a  convergent  series  where  the  separate  terms  are  holo- 
morphic functions 303 

22.3.  Proof  of  Taylor's  and  Maclaurin's  Theorems  for  functions  of 

complex  variables.     Circle  of  convergence 304 

224.  Investigation  of  the  convergency  of  varions  series  which  arc 

obtained  by  Taylor's  and  Maclaurin's  Tiieorems.     Examples  307 


inte(;i:al  calculus. 


CHArTEU  XVIII. 


KEY   TO   THE   SOLL'TIOX    OK   UIFl-KKKNTIAL   EQUATIONS. 
Article.  Page. 

226.    Description  of  Key 312 

226.  Dodnition  of  the  terms  differential  equation,  order,  degree, 

linear,  general  solution  or  complete  primitive,  singular  solu- 
tion, exact  differential  equation 012 

227.  Examples  lllnstratingr  the  use  of  the  Key 314 

228.  Sinipliflcatlfin  of  ditlerentlal  c(niatioiis  by  chan<i;e  of  variable.  ?>24 

Ki-.Y 820 

ExAMi'LEjj  T-xni-.K  Kr.Y 349 


intectEal  calculus. 


CHAPTER    I. 

SYMBOLS    OF    OPERATION. 


1.  It  is  often  convenieut  to  regard  a  runctioiial  symliol  as 
indicating  an  operation  to  be  2^(^rforme(l  upon  the  expression 
u-hich  is  ivritten  after  the  symbol.  From  this  point  of  view  the 
symbol  is  called  a  symbol  of  operation,  and  the  expression  writ- 
ten after  the  symbol  is  called  the  subject  of  the  operation. 

Thus  the  symbol  D^  in  D^{x-y)  indicates  that  the  operation  of 
dillerentiating  with  respect  to  x  is  to  be  performed  upon  tlie 
subject  (x^y). 

2.  If  the  result  of  one  operation  is  taken  as  the  subject  of  a 
second,  there  is  formed  what  is  called  a  compound  fumiiou. 

Thus  logsinx  is  a  comjMund  function,  and  we  may  s|)eak  of 
the  taking  of  the  logsin  as  a  compound  operation. 

3.  When  two  operations  are  so  related  tliat  the  compound 
operation,  in  which  the  result  of  performing  the  first  on  any 
subject  is  taken  as  the  subject  of  the  second,  U-ads  to  the  same 
result  as  the  compound  operation,  in  whidi  tlie  result  of  per- 
forming the  second  on  the  same  subject  is  taken  as  the  subject 
of  the  first,  the  two  operations  are  commutative  or  nlali eel >j  free. 

Or  to  formulate  ;  if 

fFH=Ffu. 

tlie  operations  indiratcd  l»y/  and  /-'are  <<nn mutative. 


2  INTEGRAL   CALCULUS.  [Art.  4. 

For  exainpK' ;  the  opt  riitious  of  partial  ditrcrcntiation  with 
respect  to  two  independent  variables  x  and  y  are  commutative, 
for  we  know  tliat 

l)J)^u  =  L)^D,u.  (I.  Art.  197). 

The  operations  of  taking  the  sine  and  of  taking  the  logarithm 
are  not  commutative,  for  log  sin  m  is  not  equal  to  sin  log  u. 

4.  If  f{n±v)=fa±fv 

where  u  and  v  are  any  subjects,  the  operation  /is  distributive  or 
linear. 

The  operation  indicated  I)}-  d  and  tlic  operation  indicated  b}' 
Z>,  are  distriljutive,  for  we  know  that 

d{u±  v)=dH±dv, 

and  that  A(»  ±  v)  =  D,n  ±  D^v. 

The  operation  sin  is  not  distributive,  for  sin(?<  +  v)  is  not 
equal  to  sin?t  +  sin  v. 

5.  The  comjyoxindfi  of  distributive  operations  are  distribittive. 
Let  /  and  F  indicate  distributive  operations,  then  fF  will  be 

distributive  ;  for 

F{u  ±  v)  =  Fu  ±Fv, 
tliercfore       fF{ u  ±  r)=f{ Fu  ±  Fv)  =  fFu  ±  fFv. 

<).  The  repetition  of  any  operation  is  indicated  by  writing  an 
exponent,  equal  to  the  number  of  times  the  operation  is  per- 
formed, after  the  symbol  of  the  operation. 

Thus  log'''jr  means  log  log  log  .r  ;  (J?u  means  dddu. 

In  the  single  case  of  the  trigonometric  functions  a  dilferent 
use  of  the  exponent  is  sanctioned  by  custom,  and  sin-'w  means 
(sinu)*  and  not  sin  sin?<. 

7.    if  in  and  n  are  irhole  numbers  it  is  easily  proved  that 


Chap.  I.]  SYMBOLS   OF   OlMCUATION.  3 

Thin  formula  is  assumed  for  all  values  of  m  and  n,  and  ueija- 
tive  and  fractional  exponents  are  intei-preted  by  its  aid.  It  xa 
called  the  law  of  indices. 

8.  To  find  what  interpretation  must  be  given  to  a  zero  ex- 
ponent, let  .         .      ,      .         . 

)n  =  0  HI  the  forniiila  of  Art.  7. 

or,  den'^ting/"»<  by  v\  f^v  =  v. 

That  is  ;  a  symbol  of  operation  icith  the  exponent  zero  has  no 
effect  on  the  subject,  and  ma}'  be  regarded  as  multiplying  it  by 
unit}'. 

9.  To  interpret  a  negative  exponent,  let 

m=  — n        in  the  formula  of  Art.  7. 

/-"/"«=/-«  +  »»=/'»  =  «. 

If  we  call  'fu  =  V,  then  /-"  y  =  u. 

If  v  =  l 

we  get  f~^f>t  =  "i 

and  the  exponent  —1   indicates  what  we  have  calh-d  the  anti- 
function  of /(<.      (I.    Art.  72.) 

The  exponent  —1  is  used  in  this  sense  even  with  trigonometric 
functions. 

10.  When  two  operations  are  commutative  and  disfributii'e, 
the  symbols  which  represent  them  may  be  combined  precisely  as 
if  they  were  algebraic  quantities. 

For  they  obey  the  laws, 

a(m  +  n)  =  am  +  an, 

am  =  ma, 

on  which  all  the  operations  of  arilhm.tic  and  algebra  are  fotmd«'d. 


4  INTKCKAh    CAIA'ULUS.  [Aur.  10. 

For  example;  if  the  opfnitioii  {D,+  D^)  is  to  be  performed 
n  times  in  siieeession  on  a  snlyeet  »,  we  ean  expand  (/>,  +  Z)^,)" 
precisely  as  if  it  were  a  l)in()minal,  and  then  perform  on  u  the 
operations  indicated  by  the  expanded  expression. 

(7^  +  I),y  H  =  ( /V  -f-  .'3 1)/I>^  -f-  3  D,  D;  +  Z>/)m 


i 


w.  II.]  liMAtilNAlilES. 


CHAPTER    II. 

IMAGIXARIES. 

11.  An  imaginary  is  usuall}'  defined  in  algebra  as  tJip  /»(7/- 
cated  even  root  of  a  negative  quantity^  and  although  it  is  clear 
that  there  can  be  no  quantity  that  raised  to  an  even  power  will 
be  negative,  the  assumption  is  made  that  an  iniauinary  can  be 
treated  like  an}'  algebraic  quantity. 

Imaginaries  are  first  forced  upon  our  notice  in  connection 
with  the  subject  of  quadratic  equations.  Considering  the  typical 
quadratic  ,  ,  ,   7       a 

we  find  that  it  has  two  roots,  and  that  these  roots  possess  eer- 
taiu  important  properties.  For  example  ;  their  sum  is  —a  and 
their  i^roduct  is  h.  We  are  led  to  the  conclusion  that  every 
quadratic  has  two  roots  whose  sum  and  whose  product  are 
simply  related  to  the  coefficients  of  the  equation. 

On  trial,  however,  we  fiud  that  there  are  quadratics  having 
but  one  root,  and  quadratics  having  no  root. 

P'or  example  ;  if  we  solve  the  ccjuation 

ar-2r+  1  =0, 

we  find  that  the  onh"  value  of  .»•  which  will  satisfv  it  is  unity ; 
and  if  we  attempt  to  solve 

ar-2.f4-2  =  0, 

we  find  that  there  is  no  value  of  x  which  will  satisfy  the  equation. 

As  these  results  are  apparently  inconsistent  with  tlu-  conclu- 
sion to  which  we  were  led  on  solving  the  general  equation,  we 
naturally  endeavor  to  reconcile  them  with  it. 

The  difficulty  in  the  case  of  the  cipiation  wliidi  has  hut  one 


6  intk(;hal  rAiXTLUS.  [Akt.  12. 

root  is  easily  ovcroomo  by  rc-xanling  it  as  liaviiiir  two  e(iual  roots, 
riiiis  wt-  can  say  that  oacli  of  the  two  roots  of  the  equation 

is  equal  to  1  ;  and  there  is  a  decided  advantairc  in  looking  at  the 
<lurstion  from  this  point  of  view,  for  the  roots  of  this  equation 
will  possess  the  same  properties  as  those  of  a  quadratic  having 
(ine(iual  roots.  The  sum  of  the  roots  1  and  1  is  minus  the  co- 
ellicieiit  of  X  in  the  e(|uation,  and  their  product  is  the  constant 
term. 

To  overcome  the  difliculty  presented  b}-  the  equation  which 
has  no  root  we  are  driven  to  the  conception  of  imar/inaries. 

12.  An  imcKjinary  is  not  a  quantity,  cuid  the  treatment  of 
imaginaries  is  purely  arbitrary  and  conventional.  ^Yo  begin  by 
laying  down  a  few  arbitrary  rules  for  our  imaginary  expressions 
to  obey,  which  must  not  involve  an}'  contradiction;  and  we 
must  perform  all  our  operations  upon  imaginaries,  and  must 
interpret  all  our  results  by  the  aid  of  these  rules. 

Since  imaginaries  occur  as  roots  of  equations,  they  bear  a  close 
analogy  with  ordinary  algebraic  quantities,  and  they  have  to  be 
subjected  to  the  same  operations  as  ordinary  quantities  ;  there- 
fore our  rules  ought  to  be  so  chosen  that  the  results  may  be 
comparable  with  the  results  obtained  when  we  are  dealing  with 
real  (piantities. 

13.  By  adopting  the  convention  that 

V— a-  =  a  V  — 1 , 

whore  a  is  supposed  to  be  real,  we  can  reduce  all  our  imaginary 
algebraic  expressions  to  forms  where  V  — 1  is  the  only  peculiar 
symbol.  This  symbol  V  —  1  we  shall  define  and  use  as  the  .s//»i- 
hol  of  some  opfratinu,  at  present  nnknotvn,  the  repetition  of  which, 
has  the  effect  i>f  chaufjing  the  sign  of  the  subject  of  the  op)eration. 
Thus  in  a\l  —  \  the  symbol  V  —  1  incUcates  that  an  operation 
is  perfornuMl   upon  a  which,    if  rci)eated.   will   change  the  sign 

of  a.     That  is,  

a(V-l)'-=  -a. 


Chap.  II.]  IMAGINARIES.  7 

From  this  point  of  view  it  would  be  more  natural  to  write  the 
symbol  before  instead  of  after  the  subject  on  which  it  oi)i'rates, 
(V  — 1)«  instead  of  aV  — 1,  and  this  is  sometimes  done;  but 
as  the  usage  of  mathematicians  is  overwhelmingly  in  favor  of  the 
second  form,  we  shall  employ  it,  merely  as  a  matter  of  con- 
venience, and  remembering  that  a  is  the  subject  and  the  V  — 1 
the  symbol  of  operation. 

14.  The  rules  in  accordance  with  which  we  shall  use  our  new 
sjTiibol  are,  first, 

a  V^  +  /j  V^l  =  (a  +  h)  V^l .  [1] 

In  other  words,  the  operation  indicated  by  V  —  1  is  to  be  dis- 
tributive (Art.  4)  ;   and  second, 

aV^l=(V^l)a,  [2] 

or  our  s3Tnbol  is  to  be  commutative  with  the  symbols  of  quantity 
(Art.  3). 

These  two  conventions  will  enable  us  to  use  our  symbol  in 
algebraic  operations  precisely  as  if  it  were  a  quantity  (Art.  10). 

When  no  coeflicient  is  written  before  V  —  1  the  coellicieut  1 
will  be  understood,  or  unity  will  be  regarded  as  the  subject  of 
the  operation. 

15.  Let  us  see  what  interpretation  we  can  get  for  powers  of 
V  —  1  ;  that  is,  for  repetitions  of  the  operation  indicated  by  the 
svmbol. 

(^rr-i)o=i  (Art.  8), 

(V^"l)-=  -1,  by  definition  (Art.  13), 

{yf^^iy  =  ( V"^ )- V^n  =  -  V^l ,     by  definition, 

(V^l)^=iV^n  =  v^^, 

(^/^i)«=(V'^)'^         =-1, 

and  so  on,  the  values  V— 1,  —1,  —  V— 1,  1,  occurring  in 
cycles  of  four. 


8  INTEGRAL   CALCULUS.  [AuT.  16. 

10.  The  (Icliiiition  wo  liavi-  '/iwn  for  the  square  root  of  a 
ne<i!ilive  (nuuitity,  ami  tlie  riile.s  we  liave  adopted  concerning  its 
use,  enal»le  us  to  remove  entirely  the  clillleulty  felt  in  dealing 
with  a  (|uadratie  whieh  does  not  have  n-al  roots.  Take  the 
e<iuation 

,^-_-_v  +  ;.  =  o.  (1) 

Solving  l>y  the  usual  method,  we  get 

x=l  ±  V^=a; 

V^4  =  2V^1,   by  Art.  13  [1]  ; 

hence  .r  =  1  +  "i  V  —  1   or  1  —  2  V  —  1 . 

On  substituting  these  results  in  turn  in  the  equation  (1),  per- 
forming the  o[)erations  l)y  the  aid  of  our  conventions  (Art.  14 
[1]  and  [2]),  and  interpreting  (V  — 1)"  by  Art.  15,  we  find  that 
they  botli  satisfy  the  ecpiation,  and  that  they  can  therefore  be 
regarded  as  entirely  analogous  to  real  roots.  We  find,  too,  that 
their  sum  is  2  and  that  their  product  is  5,  and  consequently  that 
they  bear  the  same  relations  to  the  eoeflicieiits  of  the  eipiation  as 
real  roots. 

17.  An  imaginary  root  of  a  (juadratic  can  always  l)e  reduced 
to  the  form  a  +  b  V  — 1  where  a  and  h  are  real,  and  this  is  taken 
as  the  general  type  of  an  imaginary  ;  and  i)art  of  our  work  will 
l>e  to  show  that  when  we  sulyect  imaginaries  to  the  ordinary 
functional  operations,  all  our  results  are  reducible  to  this  typical 
form. 

If  two  imaginaries  a-+-iV— 1  and  r-\-(l\/—{  are  equal, 
a  nuist  be  ecpud  to  r,.and  h  nuist  be  e(pial  to  tl. 

Kor  we  have  a -f  ^V— 1  =  f -|- rfV— 1. 

Tlierefore  o  —  c  =(f/  — 6)\/— 1, 

or  a  real  is  ecjual  to  an  imaginary,  unless  ii  —  ,-  =  0  =  rl  —  h. 

Sinc-e  obviously  a  real  and  au  imaginary  cannot  be  ecpial,  it 
follows  that  a  =  c  and  b  =  d. 


ClIAP.   II.] 


IMAGINAltlES. 


9 


18.  We  have  defined  V  —  1  as  the  symbol  of  an  operation 
whose  repetition  changes  the  sign  of  the  sulyect. 

Several  different  interpretations  of  this  operation  have  been 
suggested,  and  the  following  one,  in  which  every  imaginary  is 
graphically  represented  by  the  position  of  a  point  in  a  plane,  is 
commonly  adopted,  and  is  found  exceedingly  useful  in  suggest- 
ing and  interpreting  relations  between  ditl'erent  iuiaginaries  and 
between  imaginaries  and  reals. 

In  the  Calculus  of  Imaginaries,  a-\-b V—  1  is  taken  as  the 
general  symbol  of  quantit}-.  If  b  is  equal  to  zero,  a  +  b  V  — 1 
reduces  to  a,  and  is  real;  if  a  is  equal  to  zero,  o  +  6  V  —  I  re- 
duces to  6  V  —  1,  and  is  called  a  pure  imaginary. 

a  +  b-\l  —  \  is  represented  by  the  position  of  a  point  referred 
to  a  pair  of  rectangular  axes,  as  in  analytic  geometry,  a  being 
taken  as  the  abscissa  of  the 
point  and  b  as  its  ordinate. 
Thus  in  the  figure  the  position 
of  the  point  P  represents  the 
imaginary  a  +  b  V  — 1 . 

If  6  =  0,  and  our  quantity  is 

real,  P  will  lie  on  the  axis  of  

X,  wliich  on  that  account  is 
called  the  axis  of  reals  ;  if  a=0, 
and  we  have  a  jmre  imaginary, 
P  will  lie  on  the  axis  of  F, 
which  is  called  the  axis  of  j)ure  imaginaries. 

It  follows  from  Art.  17  that  if  two  imaginaries  are  equal,  the 
points  representing  them  will  coincide. 

Since  a  and  aV  — 1  are  represented  by  points  eciually  distant 
from  the  origin,  and  lying  on  the  axis  of  reals  and  the  axis  of 
pure  imaginaries  respectively,  we  may  regard  the  operation 
indicated  by  V— 1  as  causing  the  point  representing  the  subject 
of  the  operation  to  rotate  about  the  origin  through  an  angle  of 
90°.  A  repetition  of  the  operation  ought  to  cause  the  point  to 
rotate  90°  further,  and  it  does  ;  for 

and  is  represented  by  a  point  at  the  same  dist.inif  from  tin- 


10  INTEGRAL  CALCFLUS.  [Art.  19. 

nrijjin  as  a,  and  lying  on  tlic  opposite  side  of  the  origin  ;  again 
repeat  the  operation, 

and  the  point  lias  rot-ateil  '.MP  further;  repeat  again, 

and  the  point  has  rotated  through  3G0°.  We  see,  then,  that  if 
the  subjeet  is  a  real  or  a  pure  imar/iium/  the  effect  of  performing 
on  it  the  operation  indicated  by  V  —  1  is  to  rotate  it  about  the 
origin  through  the  angle  1)0°.  We  shall  see  later  that  even  when 
the  subject  is  neither  a  real  nor  a  pure  imaginar}-,  the  etlect  of 
op«'rating  on  it  with  V  —  1  is  still  to  produce  the  rotation  just 
described. 

lit.  The  .s»m,  the  product,  and  the  quotient  of  an}'  two  imagi- 
naries,  a  -|-  6  V  —  1  and  c  +  d  V  —  1 ,  are  imaginaries  of  the  typi- 
cal form. 

a  +  b^/^  +  c  +  d^T^     =a  +  c  +  (/y  +  fOV^.  [1] 

(a  +  iV^)  (c  +  d  V^ )  =  (*o  -  bd  +  {be  +  ad)'\r^.         [2] 

„-(-?,V3T  ^  (a-|-/jV3i)  (c-d^T^)  ^ ac+bd+{bc-ad)\r^ 
c  +  dyl'-i      (c+ fZ\Cr[ )  (c _ fZ V ^ )  <-•"  +  ff"' 

ac  +  bd      be  —  ad    , 

All  these  results  are  of  the  f<jnn  .1  -f  2?V— 1. 

'20.  The  graphical  representation  we  have  suggested  for 
imaginaries  suggests  a  second  typical  form  for  an  imaginary, 
(livfu  tlie  imaginary  x-f-vV  — 1,  let  the  polar  coordinates  of 
the  point  P  which  represents  x-f  y  V  — 1  be  r  and  <;^. 

r  is  called  the  modnbcs  and  <{>  the  anfuvieut  of  the  imaginary. 


CiiAP.  II.]  IMAGINARIES.  11 

The  figure  enables  us   to   establish  very 
sunple  relations  between  x,  y,  r,  and  <^. 


[•-^] 


a;=reos«^,     |  P^, 

y=?-sin<^;     j 

<^=tan-^|.  j 

3J  +  .vV— 1  =  ?'cos<^  +  (V  — l)rsin<^ 

=  r(cos<^+V^.siu</)),  [3] 

where  the  imaginar}'  is  expressed  in  tenns  of  its  modulus  and 
argument. 

The  value  of  r  given  by  our  formulas  [2]  is  ambiguous  in 
sign ;  and  <f>  may  have  any  one  of  an  infinite  number  of  values 
dirtering  by  multiples  of  tt.  In  practice  we  always  take  the 
positive  value  of  ?•,  and  a  value  of  </>  which  will  bring  the  point 
in  question  into  the  right  quadrant.  In  the  case  of  any  given 
imaginary  then,  r  can  have  but  one  value,  while  </>  may  have  any 
one  of  an  infinite  number  of  values  differing  by  multiples  of  27r. 

The  modulus  r  is  sometimes  called  the  absolute  value  of  tbe 

imaginary. 

/  Examples. 

(1)  Find  the  modulus  and  argument  of  1  ;  of  V  — 1  ;  of  —  4  ; 
of-2V^^;  of  3  +  3  V^;  of  2+4  V^;  and  express  each  of 
theso^  quantities  in  the  form  r(cos</) +V  — 1.  sin^). 

'^(2)  Show  that  ever}-  positive  real  has  the  argument  zero ; 
ever}'  negative  real  the  argument  ir ;  ever}'  positive  pure  imagi- 
nary the  argument  - ;   and  every  negative  pure  imaginary  the 

o  2 

argu;6ent  '-^. 

21.  If  we  add  two  imaginaries,  the  modulus  of  the  sum  is 
never  greater  than  the  sum  of  the  moduli  of  the  given  imagi- 
naries. 


12  INTEGRAL  CALCULUS.  [AuT.  22. 

Tln'siimof  a-f-iV^  and  c-hdV^  is  a-i-c4-(^+^0 V  — 1- 
The  modiilu.s  of  this  sum  is  y/{a  +  c)'- +  {b  +  d)'- ;  the  sum  of 
the  moduli  of  a+h^T^  and  c-f(/V^  is  V(r  +  6'+Vc=-' +  d^. 
We  wisli  to  show  tliat 


V  (a  +  c)»  +  {b  +  df  -<  V«^  +  62  +  70=^  +  0^; 
the  sign  -<  meaning  "  equal  to  or  tf'ss  than." 


Now        V(a  +  c)*  +  (6  +  d)*  -<  Va=^  + 6'  + Vc=^+ (^^ 


that  is,  if       ac  +  bd  -<  ^a'c"  +  a^d'  +  bH-  +  b^O" ; 

or,  squaring,  if 

0*0=^  +  2  abed  4-  h'd'  -<  ah'  +  a-V/-  +  b-c'  +  ^-'d^ ; 

or,  if  0  -<  (ad-bc)\ 

This  last  result  is  necessarily  true,  as  no  real  can  have  a 
square  less  than  zero  ;  hence  our  proposition  is  esta])lished. 

22.  The  vioduliis  of  the  j)^oduct  of  two  imaginaries  is  the 
product  of  the  moduli  of  the  yiven  imaginaries,  and  the  argument 
of  the  prod nrt  is  the  sum  of  the  arguments  of  the  imaginaries. 

I^t  us  multiply 

;-,(cos</», -f-V— l.sin»/)i)      by     r>(cos(/)2  +  V— 1 .  sin</)2)  ; 
we  get 

Ti  r.^[cos  </>!  cos  <^...  —  sin  0,  sin  0o -|-  V  —  1  ( sin  0,  cos  0.  +  cos  </>,  sin  </)o )], 
cos  f/),  cos  (f>.,  —  sin  </>,  sin  0^  =  cos(0,  -|-  <^._,) , 
sinc/>|Cosc/)._, -f  eoS(/),sin(^^  =  sin((^,  -f-  0^) 
by  Trigonometry  ;  hence 

r,  (cos<^,  4-V— l.siuf/),)  r,  ((•os.^j+ V^.  sin02) 
=  r,r,[cos(r^,  +  4>,)  -f-  V-i.  sin(<^,  +  0,)], 


Chap.  II.]  IMAGINARIES.  13 

;uk1  our  result  is  in  the  t^'pical  form,  fir.,  hoing  the  moduhis  and 
01  +  4>->  the  argument  of  the  product. 

If  each  lactor  has  the  modulus  unit}',  this  theorem  enaliles  us 
to  construct  ver}'  easily  the  product  of  the  imaginaries  ;  it  also 
enables  us  to  show  that  the  interpretation  of  tlie  o[)eration  V—  1, 
suggested  in  Art.  IS,  is  perfectl}'  general. 

Let  us  operate  on  any  imaginary  subject, 

r(cosc/)-f  V— 1.  sin^),  with  V^, 

that  is,  with  1  f  cos  ^  +  V^ .  sin  -  ) . 

The  modulus  r  will  be  unchanged,  the  argument  <^  will  be  in- 
creased by  ^,  and  the  effect  will  be  to  cause  the  point  rei)re- 
senting  the  given  imaginary  to  rotate  about  the  origin  through 
an  angle  of  90°. 

23.  Since  division  is  the  inverse  of  multiplication, 

?-i(cos<^i  -f  V  — 1.  sin^i)  -=-  r.2{cos<{>2+  V— l.sin<^2) 
will  be  equal  to 

-i  [cos  (<^i  -  <^2)  +  V- 1 .  sin (<^,  -  (^,)] , 

'2 

since  if  we  multiply  this  by  ?-2(cos<^2+  V— 1.  sinews)'  according 
to  the  method  established  in  Art.  22,  we  must  get 

ri(cos<^i  +  V— l.sin<^,). 

To  divide  one  imaginary  by  another,  toe  have  then  to  take  the 
quotient  obtained  by  dividing  the  modulus  of  the  Jirst  by  the 
modulus  of  the  second  as  our  required  modulus,  and  the  argu- 
ment of  the  first  viinus  the  argument  of  the  second  as  our  uetv 
argument. 

24.  If  we  are  dealing  with  the  product  of  n  equal  factors,  or, 
in  other  words,  if  we  are  raising  r(cos«^ -h  V— 1.  sin</))  to  the 


14  INTEGRAL   CALCULUS.  [Akt.  25. 

)/th  power,  ?i  licing  a  positive  whole  miinljcr,  Ave  shall  have,  by 
Alt.  22, 

[/•(cos<^-f  V^.sin0)]"  =  r"(co.s?i<^  +  -^f^ .  sin  n  (j>) .      [1] 

If  r  is  unity,  we  have  merely  to  multiply  the  argument  by  ??, 
without  changing  the  modulus  ;  so  that  in  this  case  increasing 
the  exponent  l)y  unity  amounts  to  rotating  the  point  represent- 
ing the  imaginary  through  an  angle  equal  to  <^  without  changing 
its  distance  from  the  origin. 


25.    Since  extracting  a  root  is  the  inverse  of  raising  to  a 
power, 

7[/-(cos<^  + V^.sinc/))]  =  7rfcos^-f  V^.sin-j;       [1] 

for.  bv  .\rt.  24, 


^/cos-  +  V-l.sin^j     =  r(cos<^-|- V- 


i.sin^). 


EXAMI'LK. 


•^-^how  that   Art.   24   [1]   holds  even  when  n  is  negative  or 
fractional. 

26.  As  the  modulus  of  every  quantity^  positive,  negative, 
real,  or  imaginary,  is  j)ositive^  it  is  always  possible  to  tind  the 
modulus  of  any  required  root ;  and  as  this  modulus  must  be  real 
and  positive,  //  can  ncvcr^  in  any  given  example,  have  more  than 
one  value.  We  know  from  algebra,  however,  that  every  equa- 
tion of  the  7tth  degree  containing  one  unknown  has  n  roots,  and 
that  consequently  every  number  must  have  n  nth  roots.  Our 
formula.  Art.  2;')  [1],  appears  to  give  us  but  one  j*th  root  for 
any  given  (juantity.     It  must  then  be  incomplete. 

We  have  seen  (Art.  20)  that  while  the  modulus  of  a  given 
imaginary  has  but  one  value,  its  argument  is  indeterminate  and 
may  have  anyone  of  an  infmite  number  of  values  which  dilfer  l)v 
multiples  of  2ir.     If  0u  is  one  of  these  values,  the  full  form  of 


Chap.  II.]  IMAGINARIES.  16 

the  imaginary  is  not  ?-(cos</)o  +  V^^.  sin*^,,) ,  as  we  have  hitherto 
written  it,  but  is 

r[cos(^o  +  ^^«T)  -I- V  — l.sin(</)o+2m7r)], 

where  ?n  is  zero  or  any  whole  number  positive  or  negative. 
Since  angles  ditiering  by  multiples  of  'lir  have  the  same  trigo- 
nometric functions,  it  is  easily  seen  that  the  introduction  of  the 
term  2m-n-  into  the  argument  of  an  imaginary  will  not  modify 
any  of  our  results  except  that  of  Art.  25,  which  becomes 

Vr  [cos  (00  +  -  '"tt)  +  V —T-  sin  (<^o  +  "^  '«"■)  ] 

GiAnng  m  the  values  0,  1,  2,  3  ....  ,  ?i  —  1,  n,  n  -\-\,  success- 
ively, we  get 

— 1 ■> h -^ — ' \- o —  1 \-in  —  \) — , 

n       n        n       n  n       n  n  n  n 

n  n        11 

as  arguments  of  our  ?ith  root. 

Of  these  values  the  first  h,  that  is,  all  except  the  last  two, 
coiTespond  to  different  points,  and  therefore  to  different  roots ; 
the  next  to  the  last  gives  the  same  point  as  the  first,  and  the 
last  the  same  point  as  the  second,  and  it  is  easily  seen  that  if  we 
go  on  increasing  m  we  shall  get  no  new  points.  The  same  thing 
is  true  of  negative  values  of  m . 

Hence  we  see  that  eveinj  quantity^  real  or  imaginary,  has  n 
distinct  nth  roots,  all  having  the  same  modulus,  but  with  argu- 
ments differing  by  multiples  of  ^^. 


27.  Any  positive  real  differs  from  unity  only  in  its  moddhis, 
and  any  negative  real  differs  IVom  —1  only  in  its  modulus.  All 
the  7ith  roots  of  any  number  or  of  its  negative  may  bu  obtaiued 


16 


IN  rK(;i:Ai,  cAi.crLUS. 


[Akt.  27. 


by  inulliplyiiig  the  //Ih  roots  ol'  1  or  of  —1  In  the  iv:il  positive 
7<tli  root  of  tin'  iminlicr. 

Let  us  consider  some  of  tlie  roots  of  1  and  of  —  1  ;  for  ex- 
ample, the  cube  roots  of  1  and  of  —1.  The  modulus  of  1 
is  1,  and  its  argument  is  U.    The  modulus  of  each  of  the  cube 

roots  of  1  is  1 .  and  their  arguments  are  0,  — ,  and  —  ;  that  is, 

o  o 

0°,   120°,   and  240°.     The  roots  in  question,  then,  are  repre- 
sented by  tlie  points  7*i,  P.,  I\s  in  the  figure.     Their  values  are 


l(cosO-f- V^.  sinO), 
1  (cos  120°  +  V^.  sin  120°), 
and  1  (cos  240° +  V^.  sin  240°), 
or  1,  _^  +  f  V^l,    _^_-^V^. 


The  modulus  of 
argument  is  it.     Th 


1   is  1,  and  its 
modulus  of  the 


cube  roots  of 

47r 


1   is   1 


and  their  arguments  are  -,  -  -I , 

3    3        3 


^  +  -^,  that  is,  G0°,   l.SO°,  300°.     The  roots  in  question,  then, 
o        3 

are  represented  by  the  points  I\,  P.,, 

1\,   in  the  figure.      Their  values  are 


i  +  f  V-1,  -1,  \ 


V=T. 


Examples. 

(1)  "What  are  the  square  roots  of 
1  and  -  1  ?  the  4th  roots  ?  the  5th 
roots  ?   the  6th  roots  ? 


(2)    Find  tlir  (Mihr  roots  of  -s  ;   the  .^)tii  roots  of  32. 


(3)  Sliow  th.Mf  :iu  imaginary  can  have  no  real  »ith  root:  that 
a  imsitive  real  has  two  real  »th  roots  if  n  is  even,  one  if  n  is 
od.l :  tliMt  :i  m-Mtivr  ival  has  one  real  ),th  root  if  «  is  odd,  none 
if  //  is  i'\(ii 


Chap.   II. j  IMA<;iNAUIES.  17 

28.  Imaginarios  having  equal  moduli,  and  arguuHiits  ditli'iinii 
only  in  sign,  are  called  co)iJi((jn(e  iiiKKjiiutries. 

r(cos<^  + V— l.sin<^),  and  r[cos(  — 0)  +  V— l..siii(  —  ,^)]. 
or  r(eos^  —  V—  1 .  sin  <f))  are  conJiH/ate. 

They  can  be  written  a;  +  y  V  —  1  and  x*  —  ?/  V  —  1 .  and  we  see 
that  the  points  corresponding  to  them  have  the  .same  abscissa, 
and  ordinates  which  are  equal  with  opposite  signs. 

Examples. 

1^1)  Prove  that  CO ?y«9a/e  imaginariea  have  a  real  sum  and  a 
real  product. 

)u!)  Prove,  by  considering  in  detail  the  sul)stitution  of 
o  -f-^V— 1  and  a  — i»V— 1  in  turn  for  x  in  any  algebraic  poly- 
nomial in  X  with  real  coefficients,  that  if  any  algebraic  equation 
witli  real  coefficients  has  an  imaginary  root  the  conjugate  of  that 
root  is  also  a  root  of  the  equation. 


Prove  that  if  in  any  fraction  where  the  numerator  and 
denominator  are  rational  algebraic  polynomials  in  .r,  we  substi- 
tute «-f-^V— 1  and  a  — iV^n  in  turn  for  a;,  the  results  are 
conjugate. 

Transcendental  Functions  of  Imaginaries. 

29.  We  have  adopted  a  definition  of  an  imaginary  and  laid 
down  rules  to  govern  its  use,  that  enable  us  to  deal  witli  it,  in 
all  expressions  involving  only  algebraic  operations,  precisely  as 
if  it  were  a  quantity.  If  we  are  going  further,  and  are  to  sub- 
ject it  to  transcendental  operations,  we  must  carefully  define 
each  function  that  we  are  going  to  use,  and  establish  the  rules 
which  the  function  must  obey. 

The  principal  transcendentid  functions  are  c^,  log.)",  and  sin./-, 
and  we  wish  to  define  and  study  these  when  .i-  is  replaced  liy  an 
imaginary  variable  z. 

As  our  conception  and  treatment  of  imaginaries  have  been 
entirely  algebraic,  we  naturally  wish  to  define  our  transcentlental 


18  INTEGRAL    CALCULUS.  [Art.  30. 

functions  by  the  aid  of  ulgobraic  functions ;  and  since  wc  know 
that  the  tran.sccndcntal  functions  of  a  real  variable  can  be  ex- 
pressed in  terms  of  algebraic  functions  only  by  the  aid  of  infinite 
series,  we  are  led  to  use  such  series  in  defining  transcendental 
functions  of  an  iimujinary  variable  ;  but  we  must  first  establish 
a  i)roposition  concerning  the  convcrgeucy  of  a  series  containing 
imaginary  terms. 

30,  If  the  moduli  of  the  terms  of  a  series  containing  imaginary 
terms  form  a  convergent  series,  the  given  series  is  convergent. 

Let  ?/o  +  "i  +  "a  + +  "n  + IJL'  '-^  series  containing  imagi- 
nary terms. 

Let  

«o=  i?o(cos<I>o+  V— l.sin<l>o),  111  =  ^i(eos^i-f- V— l.sin^i),  (fee, 

and  suppose  that  the   series  Iio+Iii+R2  + +  ^^n  + is 

convergent ;  then  will  the  series  Mq-I-  "i+  ^h-\- be  convergent. 

The  series  JiQ-\-  Iii-\- is  a  convergent  series  composed  of 

positive  terms  ;  if  then  we  break  up  this  series  into  parts  in  any 
way,  each  part  will  have  a  definite  sum  or  will  approach  a  defi- 
nite limit  as  the  number  of  terms  considered  is  increased  in- 
definitely. 

The  series  Wq -f- », -|- "^ -f- "„ -f can  l)e  broken  up  into 

the  two  series 

i?ocos*o-|-  7?,cos<I>,  -I-  7^oCos*2  + +  -'?,.fos<I>„-|- (1) 

and 

^/^{Ros\n%  +  RiHin^i  +  R.jsm<i>2 -\- +  /i'„sin*„-|- ).  (2) 

(1)  can  be  separated  into  two  parts,  the  first  made  up  only 
of  positive  terms,  the  second  onl>'  of  negative  terms,  and  can 
therefore  be  regarded  a.s  the  difference  between  two  series,  each 
consisting  of  positive  terms.     Each  term  in  either  series  will  be 

a  term  of  the  modulus  series  li^  +  Ri  -\-  R2  + multiplied  by 

a  quantity  less  than  one,  and  the  sum  of  n  terms  of  each  series 
will  therefore  approach  a  definite  limit,  as  n  increases  indefi- 
nitely. The  series  (1),  then,  which  is  the  abscissa  of  the  point 
representing  the  given  imaginary  series,  has  a  finite  sum. 


Chap.  II.]  IMAGIXARIKS.  19 

In  the  same  wa}-  it  ma^-  l>e  shown  tliat  the  cocfrieieiit  of  V— 1 
in  (2)  has  a  finite  sum,  and  this  is  the  ordinate  of  the  point 
representing  the  given  series.  The  sum  of  n  terms  of  tlie  given 
series,  then,  approaches  a  definite  Umit  as  ?i  is  increased  indefi- 
nitely, and  the  series  is  convergent. 

31.    We  have  seen  (I.  Art.  133  [2])  that 

^  =  ^+I  +  ll  +  i7  +  l7  + f] 

when  X  is  real,  and  that  this  series  is  convergent  for  all  values  of  .r. 
Let  us  define  e',  where  z  =  x-j- y^f  —  l,  by  the  series 

«'=i+Y+|i+|^+|^+ m 

This  series  is  convergent,  for  if  ^  =  /-(cos  <^  +  V  —  1 .  sin  <^)  the 
series  ^ 

1+-+— +— +— + 

i2!3:4: 

made  up  of  the  moduli  of  the  terms  of  [2]  is  convergent  by 
I.  Art.  133,  and  therefore  the  value  we  have  chosen  for  e'  is  a 
determinate  finite  one. 

Write  x-\-y\/—l  for  z,  and  we  get 

c^+vv^^l  I  •^•+W^l  I  (■x-+W~l)-  ,  (a-+.W~l)'  I  |-3-| 

The  terms  of  this  series  can  be  expanded  by  the  Binuiiiial 
Theorem.  Consider  all  the  resulting  terms  containing  any  given 
power  of  X,  say  x^ ;  we  have 

^'  n  I  ^'^^^  I  (!'^~^y-  I  (W~i)'^  I        ,  (.yV^n", y 

pr  ^  1  ^  2!  "^  3!  ^  ^  u\  ^  " 
or,  separating  the  real  terms  and  the  imaginary  terms, 

j>r      2!    4!    g:        ' 

p\  3 !       o !       /  ! 


20  INTi:(iKAL    CALCULUS.  [AUT.  32. 

or  —  (cos v+ V^. sill'/).  hv  I.  Art.  134. 

P- 

Giving  ])  all  valiu-s  IVoiii  (»  to  x  wc  get 

e'^»-'^=(cosy  +  V-l.siu//)(l  +  Y  +  ^  +  |^  +  j^+ ) 

=  c'  ( cos  V  +  V  —  1 .  sin  V ) ,  [4] 

which.  In-  the  way.  is  in  one  of  our  typical  iiuas^inary  Ibrnis. 

If..=  0.   in  [4], 
we  get  (-y  v^->  =  cos »/  +  V—  1 .  sin  ?/,  [,'»] 

which  suggests  a.  new  way  of  writing  our  typical  iniaginar}' ; 
namely,  

r(cos(f>  +  V— I.  sin<^)  =  ir'^'K 

.32.    We  have  seen  th.nt 

let  us  see  if  all  iniagiiuiry  powers  of  e  obey  the  law  of  indices; 
that  is,  if  the  equation 

e"e''=e''  +  '  [1] 

is  universally  true. 

Let  n  =  .<-,  +  7,  V—  1     and     r  =  .7V 4- 1/.^  V  —  1 , 

then  <»"=  e'l  +  Wi  ^-'  =  f^i (cos?/,  -|-  V—  1 .  sin  Vi) , 
g. ^  px,  +  y,  v/TT  _  ,,xj(cos ?/2  +  V—  1 .  sin//.,) , 

e"e''  =  e^ic^j  [cos (,v, +  //,,)  +  V— 1.  sin  (?/,  + ?/,,)] 
=  e*.  +  ^-^  [cos (//,  -f-  ?/,)  +  V^ .  sin {>/,  +  //,) ] 


an<l  tlie  fitiuldini'vtnl  proju'rtt/  of  exponential  functions  holds  for 
imatfiuaricti  as  tri'll  its  fhr  nals. 


I 


KXAMIM.K. 

'rove  that  <»"«'  =  (("  +  ''  when  yi  and  /"  are  imaginary, 


CiiAI'.   II.]  IMA(ilNAliIES.  21 

Logo  n't  It  m  ic  Fu  iictio  ns. 

33.  As  a  loiiaritlun  is  the  iiivtTse  of  an  expoiK'iitial.  we  oii^lit 
to  be  able  to  obtain  the  logarithm  of  an  imaginarv  from  the 
formula  for  e"""^*^"'.    We  see  readily  that 

z  =  r  (cos</,  +  V^.  siu0)  =  t'^^sr+o^'-i^ 
whence  logz  —  log  r  +  c/)  V  —  1  ; 

or,  more  strictly,  since 

z=  ?'[cos  (^0  +  2"7r)  +  V— l.sin(<^u+  2»7r)], 

log2  =  log/-+ (</)o+2n7r)  V^  [1] 

Avhere  ii  is  an}'  integer. 

If  z  =  .r  4-  Z/  V^,  r  =  V.t;-4-/,  and  </>  =  taii"'?-^ ; 

X 

whence  logz  =  4^ log  (.r  +  f)  +  V  —  1 .  tan"'  ^.  [2] 

Each  of  the  expressions  for  log  z  is  indeterminate,  and  repre- 
sents an  intinite  number  of  values,  dilfering  l)y  multiples  of 
27rV^. 

This  indeterminateness  in  the  logarithm  might  have  l)een  ex- 
pected a  jyriori,  for 


=  cos  Z  TT 


+V-l.sin2  7r=  1,       by  Art.  31 


Hence,  adding  2  7rV— 1  to  the  logarithm  of  any  (piantity  will 
have  the  effect  of  multiplying  the  quantity  by  1,  and  therefore 
will  not  change  its  value. 

Example. 

Show  that  if  an  expression  is  imaginary,  all  it,s  logarithms  are 
imaginary  ;  if  it  is  real  and  positive,  one  logarithm  is  real  and 
the  rest  imaginary  ;  if  it  is  real  and  negative,  all  are  inuigiiuiry. 


^^  INTKGKAL    (ALCrLUS.  [Aur.  34. 


Trifjonometric  Fidictions. 
34.    n-isn-al, 

Z^  T"  y"! 


I'vl.  Art.  l;;i.  ^I      4!      G 


.'4!      G!^  M 


tiR'  .scries  of  the  moduli, 


/-■•         <-'         ..7 


y-  *•■'  .•« 

2  !       4  !  ^  G  !  ^ 

se I.e.  [1]  and  [2]  are  eovergent.     We  .shall  take  them  as  defi- 
nitions of  the  sine  and  cosine  of  an  imajrinaiy. 

KXAMI'I.K. 

^    From  th(.  formulas  of  Art.  ;51,  and  from  Art.  34  \ 
show  that  '- 


1]  and  [2], 
smz, 

sin^,     for  all  values  of  «. 
35.    From  the  relations 

<^''^"'  =  ^'OS2  +  V^.  sin  2, 
e-'^'  =eos2-  V^..sin2, 

2  '  [1] 

Sm2= ^  r-,^ 

Tor  all  values  of^.  -  v      i 


Chap.  II.] 
Let 

cos(.r-}-?/V— 1)  = 


IMAGINARIES. 


z  =  x  +  y\J  —  \. 


23 


+  e- 


■.^/^l  +  y 


_  (cos.r+  V—  1 .  sina;)e~''+  (cos  a;—  V—  1 .  sin  a;)e* 
2 

by  Art.  34,  Ex., 


=  cosx  — ■ —  V  — l.smx' 


[3] 


In  the  same  wa^'  it  niav  be  shown  that 


.     .     ^1 — rN  _(cos.x--f- V— 1.  sina;)e  »'— (cosa;  —  V  — l.sina;)( 

H^x+j/ V  — i;— -—iz 

2V-1 


e"  +  e- 


+  V-1. 


e^  -  e-*' 


If  2  is  real  in  [1]  and  [2],  we  have 


cos  X  = 


+  e- 


[i] 


,a,.  =  _l ^ V-1. 


If  z=7/  V  — 1,  and  is  a  pure  imaginar}', 

e*  +  e- 


1  = 


sm?/ 


V^  = 


V:^!; 


[5] 
[6] 


whence  we  see  that  the  cosine  of  a  pure  imaginary  is  real,  while 
its  sine  is  imaginary. 

By  the  aid  of  [5]  and  [<i],  [3]  and  [4]  can  be  written  : 

cos  (ic  4-  ?/  V  —  1 )  =  cosx'cosyV— 1  —  sinxsin// V— 1,     [7] 
sin  (x  + >/ V— 1)  =  sinx-cosy  V^  -I-  cosxsin// V— 1.     [h] 


24 


LNTKiiKAL   CALCl'LUS. 


[Akt.  36. 


EXAMI'LES. 


(;)    From  [1]  and  [2]  sIk.w  Uiat  siir2+ cos^j;  =  1. 

1/(2)    rrovi-  tliat 

cos  ( II  -f  v)  =  COS  a  t'O.s  V  —  sin  n  sin  /', 
sill  {n  +  v)  —  sin?<cosi'  +l'OS?(sin«', 

where  n  and  r  arc  iinaginarv. 

The  rchitions  to  lu'  proved  in  examples  (1)  and  (2)  are  the 
Ciindanicntal  rorniuhis  of  Trigonometry,  and  they  enable  ns  to 
use  trigonometric  functions  of  iinaginarics  precisely  as  we  use 
tritionoinetric  functions  of  reals. 


Differentiation  of  Functions  of  Inuajinaries. 
36.    A  function  of  an  imaginary  variable, 

is,  strictly  speaking,  a  function  of  two  independent  varial)les, 
.(•  and  // ;  for  we  can  changt'  z  by  changing  either  x  or  y,  or  l)oth 
./•  and  _'/•  Its  differential  will  usually  contain  dx  and  chj,  and  not 
necessarily  dz ;   and  if  we  divide  its  dilferential  by  Oz  to  get  its 

derivative  with  respect  to  z,  the  result  will  generallv  contain  -;^. 

dx 

which  will  be  wliftliy  indeterminate,  since  x  and  y  are  entirely 
independent  in  the  exprt'ssion  x  -f- .'/  V  —  1  •  It  may  happen, 
however,  in  the  case  of  some  simple  functions,  that  dz  will  appear 
MS  a  factor  in  the  differential  of  the  function,  which  in  that  case 
will  have  a  single  derivative. 

.'57.  ///  dijTirfnfidtiuij.  the  V^l  may  he  treated  tike  a  con- 
stant;  for  the  operation  of  finding  the  dilferential  of  a  fimction 
is  an  algebraic  operation,  and  in  all  algebraic  operations  V  — 1 
obeys  the  same  laws  as  anv  consfant. 


[AP.    II.] 


IMAG  IN  ARIES. 


Example. 
Prove  that  rf(ar V^)  =  2  a;  V^ .  dx ; 

and  that  d  V— 1 .  sin .r  =  V  —  1 .  cos x.dx. 

We  have,  by  the  aid  of  this  i)iinciple, 
if  z=x-i-y^/^, 

dz  =  dx  +  \r^.dy;  [1] 

if        2  =  r(cos(^  +  V— 1.  sin^), 

dz  =  dv{cos(f>  +  V  — 1.  sin<^)  +  rdcf){—  sin  </>  +V—  1.  cos  eft) 
=  {dr  +  rV^.dcji)  (cos </>  +  V^l .  sin  <^) .  [2] 

38.    Let  ns  now  consider  the  dilferentiation  of  2",  e',  logz, 
sinz,  and  cosz. 

Let  z=  r  (cos <^  +  V  —  1 .  sin (/>) , 

then 
2"*  =  r'"(cosm<^  +  V^.  sin?u^),  by  Art.  24  [1]  ; 

dz'"  =  ?»)•"'"' r/;- (cos ?«(^  +  V— 1.  sin??i^)  +  mr^dcf:  {—  sin  ?n  (/> 

+  V  — l.COS/«(^), 

^2"=  ?«)•"'"'  [cos  (»i  — 1)  <^  +  V  —  1.  sin (/»  —  !)  <^]  (cos(^ 
+  V^.sinf/))f//- 

+  ?» /•"•  [cos  ( ?»  —  1 )  <j!)  +  V  —  1 .  sin  (?»  —  1)0]  (cos <f> 
+  V^ .  sin  (^)  V^ .  (7<^, 

dz"  =  ?>(?•'""'  [cos  (?H  — 1)  <^  4-  V— l.sin  (m— 1)  <^]  {dr 

+  rV  — 1  .cZ(^)  (cos</>+  V— 1.  sin0). 
dz'"  =  mz"'-' dz,  [  1  ]     1  'V  Art .  3 7  [2] , 


dz 


[■^1 


and  a  power  of  :in  imaginar}'  varial)le  has  a  single  di-rivativc. 


26  INTEGRAL   CALCULUS.  [Art.  39. 

39.    If     2=a;-|-t/V^, 

e'  =  e'(cos2/+V^..sin?/),  bj- Art.  31  [4j, 

de'  =  e'c/x(cosy-f-V— 1.  sin?/)  +  (f{—  ainy 
+  V^.cos?/)f7?/, 

de*  =  e*(cosy+'\/  —  l.  siny)  (cte+V^.  cly), 
de'=e'dz,  [1] 


dz 
Show  that 


[2] 


Example. 

da'  =  a'  lo<ir  a.dz. 


\a^  ai 


Ki^'^'-l 


40.    If  2  =  »-(co.s<^+V3T.si„^), 

log2;  =  log)-  +  <^V^, 

r  r 


sJ^.-^- 


by  Art.  33, 


rflogz  = 


(r^--frV— I.d0)(cos<^+V-I..sin<^) 


•(cos  <^  +  V—  1 .  .sin  <^) 


dlog2  =  — , 

z 

dlogz_  1 
dz         z' 


[1] 


sinz 

= 

e' 

^^i_e-'^' 

2V-n 

fLsinz 

= 

p' 

v^-.  +  ,>-^i 

2  V^ 

e* 

^i_^,-.^--I 

V-l.rfz 


tfe, 


dsinz  =  cosz.rfz. 


by  Art.  35  [2], 


by  Art.  35  [1], 
[1] 


cosz  = ■ 


CiiAP.  II.]  IMAGINARIES.  27  

rfC0S2  =  ; V  — l.rfz  = — ffe, 

f?cos2;=  —  sinz.dz.                                                           [2]  I 

I 

42.  AYe  see,  then,  that  we  get  the  same  formulas  for  the  dif- 
ferentiation of  simple  functions  of  imaginaries  as  for  the  dif-  \ 
ferentiation  of  the  corresponding  functions  of  reals.     It  follows  i 
that  our  formulas  for  du-ect  integration  (I.  Art.  74)  hold  when  x  \ 
is  imaginar}'. 

HyperhoUc  Functions. 

43.  We  have  (Art.  35  [.5]  and  [H])  j 


cos  X 


V3i=£!±r 


and  sin.rV^  =  ^^^V-l, 

where  x  is  real.     — ^^— —  is  called  the  hyperbolic  cosine  of  «, 


e^-e 


and  is  written  cosh.r;  and  — ^ —  is  called  the  hyperbolic  sine 
of  .r,  and  is  written  sinh.r; 

sinhx  =^-^1^-^  =  —  V  — l.sin.x^V  — 1,  [1] 

cosh.r= — ^t— -  =  cos.r  V— 1.  [2] 

The  hyi)erbolic  tangent  is  defined  as  the  ratio  of  sinh  to  cosh  ; 
and  the  hyperbolic  cotangent,  secant,  and  cosecant  are  the  re- 
ciprocals of  the  tanh,  cosli,  and  sinh  respectively. 

These  functions,  which  are  real  when  x  is  real,  resemble  in 
their  properties  the  ordinary  trigonometric  functions. 


^ 


28  INTEGRAL    CALCULUS.  [Art.  44. 

44.    For  example, 

eoslr  .>•  —  .siiil)-.r  =  1  ;  [1] 

p^^  +  2  +  e-^' 

for  cosh- a;  =  — -- — , 

4 

and  sinh^'a;  = ^^t_? 

4 

Examples. 
(  (1)   Prove  that         1  —  tauh'-.i-  =  sech'a;. 

(2)   Prove  that         1  —  ctnh^.r  =  —  csch^a;. 
i  (3)   Prove  that       sinh(x +  _?/)  =  siiih.x* cosh?/ +  eoshxsinhy. 
\  (4)   Prove  that       cosh (x +  ?/)  =  cosh x cosh //  + siuh a- siuhy. 


45.  rtsinh.r  =  « = — ' ax. 

2  2 

rfsiuhx=  coshx.dx. 


Examples. 
( 1 )   Prove         1    d cosh  x  =  sinli  x.dx. 
\  dtanhx  =  scclvx.dx. 
\  dctuhx  =  —  csch^x.c/a;. 
^  1  d  sech  x  =  —  sech  .r  tanh  x.dx. 

*\   dcschx  =  —  cschxctiih x.f/x. 

46.    We  can  deal  with  anti-hyperbolic  functions  just  as  with 
anti-trijiononietric  functions. 
To  find  f/sinh  'x. 

Let  «  =  siuh  'x, 

then  x  =  8inhj<, 

dx  =  cosh  ii.dii. 


IP.  U.j                                INIAOIN ARIES. 

29 

cosnw 

cosh?f  =  Vl  +siuh^M, 

by  Art.  44  [IJ, 

cosh»=  Vl  +af^, 

7                11                                   (^X 

ttsinh    X  = • 

ri] 

Vl+ar' 

EXAMI'I.KS. 

Prove  the  formulas 

11              <^^'^' 

Vo;-  —  1 

^      dtanh^a;  =  - —> 

I       ■,       11                    (Jx 

x^/l-x" 

I     dcsch^a;  = — 

x  yxr  +  1 

47.    The  anti-hyperbolic   funetious  are  easily  expressed  as 
logarithms. 

Let  w  =  sinh-'.r, 


then 


2x=  e" , 


2xe"  =  e'-"—  1, 
e^"  — 2.te"=  1, 
e2"-2a-e"  +  .vr=  1  +  ar'. 


e"  =  a-  ±  Vl  +a^; 


30  INTEGRAL   CALCULUS.  [Art.  48. 

as  e"  is  necess.irily  positive,  we  may  reject  the  negative  value  in 
the  second  iueiul)er  as  impossible,  and  we  have 


=  a;+ Vl+ar^, 


n  =  log(a;  +  VT+l?), 


Examples. 
Prove  the  formulas 


»  /cosh"'x=log(a;+  Vx^—  1). 

,  .-tauh ^x  =  ^ log  . 

*^  '        1  —X 


48.  One  of  the  advantages  arising  from  the  use  of  hyper- 
bolic functions  is  that  they  bring  to  light  some  curious  analogies 
between  the  integrals  of  certain  irrational  functions. 

From  I.  Art.  71  we  obtain  the  formulas  for  direct  integration. 

j — '-==^i:_    =  siu'^x.  [1] 

J  Vl  —  X- 

f — =:sec-'a;.  p]] 

From  Art.  4G  we  obtain  the  allied  fonnulas : 

/— ^  — =sinh  'x  =  lo?(.r4- Vl-f^').  m 

f — ^  ^       =  cosh  '  X  =  log(.i-  +  Var  —  1 ) .  [5] 


CiiAP.  IT]                               BfAGIN ARIES. 

81 

J  1  -  ar                                -        1  -X 

[6] 

r     dx      __„..-, „_,_/!  ,    IT 

-) 

[-] 

JxVl-a^                           °\^      ^^' 

C        ^^-^               c-ch-^x       \0"f^    1    u    ^ 

-)• 

[8J 

J«Var  +  l                           "U      \x- 

Examples. 

Prove  the  formulas 

^m    cinh.r  =  — -^"^-4-  — -^ 

^(2)  coshx  =  l  +  |j  +  |j  + 

t/(3)  sin  (x-\-  1/  V—  1)    =  sin  x  cosh  ?/  +  V—  1  cosx  sinh  y. 

i/(4)  cos(x  +  yV— 1)    =  cos  X  cosh  y — V—1  sin  X  sinh  y. 

/KN  4.      /I        / — Tx        sin  2x  +  V—  1  sinli  2y 

(5)  tan(x  +  W-l)   -       cos2x  +  cosh2>/       ' 

(6)  sinh  (x  -f-  y  V—  1)  =  sinh  x  cos  y  +  V—  1  cosh  x  sin  ?/. 

(7)  cosh  (a;  +  >j  V—  1)  =  coshx  cosy  +  V—  1  sinhx  siny. 

/ON  ^     w     I        /~Tx       sinh  2x  +  V—  1  sin  2y 

(8)  tanh  (x  +  y  V— 1)  = t—^ — ; n — 

^  ^  V     '  ^  /  cQsh  2a;  +  cos  2y 


(9)    tanh  ^x 


=  ^+3  +  5  + 


INTEGRAL    CALCULUS.  [AUT.  4'J. 


CHAPTER    TIT. 

QENEIIAL    METHODS    OF    INTEGRATING. 

49,  We  have  dclinod  tlio  integral  of  an}-  function  of  a  single 
variable  .as  the  function  which  has  the  given  function  for  its 
derivative  (I.  Art.  53)  ;  we  have  defined  a  definite  integral  as 
the  limit  of  the  sum  of  a  set  of  differentials ;  and  we  have  shown 
that  a  definite  integral  is  the  difference  between  two  values  of  an 
ordinary  integral  (I.  Art.  1H3). 

Now  that  we  have  adopted  tiie  ditferential  notation  in  place  of 
the  derivative  notation,  it  is  better  to  regard  an  integral  as  the 
inverse  of  a  differential  instead  of  as  the  inverse  of  a  derivative. 
Hence  the  integral  of  fr.dx  will  be  the  funct^pn  whose  differ- 
ential \s  fx.d.r  ;  and  we  .shallindicatc  it  b}'  jfx.dx.  In  our  old 
nt>tati()ii  we  should  liave  indicated  precisely  the  same  function  by 
I  /> ;  for  if  the  derivative  of  a  function  is  fx  we  know  that  its 
dilli-rential  in  fx.dx. 

60.  li  fx  is  a  continuous  function  of  .r,  fx.dx  has  aji  integral. 
For  if  we  construct  the  curve  whose  equation  is  y  =  fx,  we  know 
that  the  area  included  by  the  curve,  the  axis  of  X,  an}-  fixed 
ordinate,  and  the  ordinate  corresponding  to  the  v.ariable  x,  has 
for  its  ditrerential  ydx,  or,  in  other  words,  fx.dx  (I.  Art. ')1). 
Such  an  area  always  exists,  and  it  is  a  detennin.ate  function  of  .r, 
except  that,  as  the  position  of  tiie  initial  ordinate  is  wholly  arl)i- 
trary,  the  expression  for  the  area  will  contain  an  arbitrary  con- 
stant. Thus,  if  Fx  is  tlie  area  in  question  for  some  one  position 
of  the  initial  ordinate,  we  shall  liave 

ffx.dx=  Fx+C, 

where  C  is  an  arl)itrarv  constant. 


H(ji  -F,'F[i>] 


Chap.  III.]  GENERAL   METHODS   OF  INTEGRATING.  33 

Moreover,  Fx-\-C  is  a  complete  expression  for  \  fx.dx  ;  for  if 
two  functions  of  .r  have  the  same  differential,  the}^  have  the  same 
derivative  with  respect  to  x,  and  therefore  they  change  at  the 
same  rate  when  x  changes  (I.  Art.  38)  ;  they  can  ditfer,  then, 
at  any  instant  only  by  the  difference  between  their  initial  values, 
which  is  some  constant. 

Hence  we  see  that  every  expression  of  the  form  fx.dx  has  an 
integral,  and,  except  for  the  presence  of  an  arbitrary  constant, 
but  one  integral. 

51.  We  have  shown  in  T.  Art.  18,3  that  a  definite  integral 
is  the  ditference  between  two  values  of  an  ordinary  integi'al,  and 
therefore  contains  no  constant.  Tluis,  if  Fx  +  C  is  the  integral 
of  fx.dx, 

'^' fx.dx  =  Fb-  Fa. 


£■ 


In  the  same  way  we  shall  have 

/z.c?2  =  Fb  -  Fa 


£ 


and  we  see  that  a  definite  integral  is  a  function  of  the  values 
beticeen  xchich  the  sum  is  taken  and  not  of  the  variable  with 
respect  to  which  we  integrate. 


Since 


C''fr.dx=Fa-Fb, 
C" fx.dx  =  —  C  fx.dx. 

Example.  /    //  ^i   -  /    <-  "' 


fx.dx  +  r fx.dx  =  i  fx.dx.      ,  u 

52.  In  what  we  have  said  concerning  definite  integrals  we 
have  tacitly  assumed  that  the  integral  is  a  continuous  function 
between  the  values  l)etween  which  the  sum  in  (juestion  is  taken. 
If  it  is  not,  we  cannot  regard  the  whole  increment  of  Fx  as  ctiual 


/-^' 


34  INTKGIIAL   CALCULUS.  [Art.  53. 

to  the  limit  of  the  sum  of  the  partial  infinitesimal  increments, 
and  the  reasoning  of  1.  Art.  183  ceases  to  be  valid. 
Take,  for  example.    ("  ^.    ^T-ll '  -   -'- 


\lx 


j?=j 


- 1 


i,     by  I.  Art.  55  (7) 


and  apparently 


J-i  x~      \     xj^^i      \     xj^^_i 
I     -^  ought  to  be  the  area  between  the  curve  w  =  — ,  the 


But 


axis  of  X,  and  the  ordinates  corresponding  to  a;  =  1  and  x  =  —  1 , 

which  evidently  is  not  —2  ;  and  we 
1 


see  that  the  function  —  is  discon- 

or 
tinuous  between  the  values  x=  —1 
and  x  =  l. 

The  area  in  question  which  the 
definite  intogi'al  should  represent  is 
easily  seen  to  be  infinite,  for 


J- 1     rr      € 


1, 


and  each  of  these  expressions  increases  without  limit  as  e  ap- 
proaches zero. 


58.  Since  a  definite  integral  is  the  difference  between  two 
values  of  an  indefinite  niiegral,  what  we  have  to  find  first  in  any 
problem  is  the  indefinite  integral.  This  may  be  found  by  in- 
spection if  the  function  to  be  integrated  comes  under  an}'  of  the 
forms  we  have  already  obtained  by  differentiation,  and  we  are 
tlien  said  to  integrate  directly.  Direct  integration  has  been  illus- 
trated, and  the  most  important  of  the  forms  which  can  be  in- 
tegrated directly  have  l)een  given  in  I.  Chapter  V.  For  the  sake 
of  convenience  we  rewrite  these  forms,  using  the  differential 
notation,  and  adding  one  or  two  new  forms  from  our  sections  on 
hyperbolic  functions. 


-^ 


Chap.  III.]     GENERAL   METHODS   OF    INTEGRATING.  30 

2..-/-  =  log- 

:5^Ca'dx  =  -^. 
^      J  log  a 

^T^\  sin.r.(?x=  — cosa;. 
^•H  cos.r.fZa;  =  sinx*. 
'^— I  tan.x-.f?.c  =  —  log  cos  a;. 

S--j  ctn  a-,  f/a-  =  log  sin  x. 

qS     '^^       =sin-^r. 

^^r  ^-^    =si'iA^r = iog(.v + vr+1?). 


d.r 


Vx-2-1 


=  cosh  ^T  =  log  (X  +  Va-^  —  1 ) . 


+  .X-' 


.  ^    r   ^^•^'     =taub  ^T  =  ilog{-^^. 
/  ^^  l-ar  1  -a; 

,^  r— ^g =  vers  'x. 

//J  V2X-X2 


36  INTEGRAL  CALCULUS.  [Art.  64. 

54.  "We  took  up  in  I.  Chap.  V.  the  principal  devices  used  in 
preparing  a  function  for  integration  when  it  cannot  be  integrated 
directly. 

The  first  of  these  methods,  that  of  intefjration  by  siibstitutmi, 
is  simplified  by  the  use  of  the  ditlerential  notation,  because  the 
formula  for  change  of  variable  (I.  Art.  75  [1]), 

j  /<  =  I  uD^    becoming    |  ridx=  i  u—dy, 

reduces  to  an  identity  and  is  no  longer  needed,  and  all  that  is 
required  is  a  simple  substitution. 

(a)   For  example,  let  us  find  |  ^Vl-f  logx. 
Let     1  +  logx  =  z  ;  then     —  =  dz, 


and 


r^  Vl  +  l()g.C  =    Cz^dz  =  2  2^  =  2  (1  _^  log.'C)^ 


When,  as  in  this  example,  a  factor  of  tlu  quantity  to  be 
integrated  is  equal  or  proportional  to  the  differential  of  some 
function  occurring  in  the  expression,  tlie  sul)stitutiou  of  a  new 
variable  for  the  function  in  question  will  generally  simplify  the 
problem. 

(b)  Itcjuircd  f— ^^. 

Let     e'  =  y\  then     e'dx  =  dy, 

dx      _    e'dx  __     dy 
e'  +  i'-'  ~  e-^ +T  ~  y-  +  l' 

and  I  —^ —  =  I     '  •'  .,  =  tan  '  i/  =  tan  V. 

(c)  Re(piiri'<l  I  secx.^ 


dx. 


J coso; 

cos.r      cos'' a; 


Cii.vi'.  III. J     GENERAL   METHODS    OF    INI^mm^UATING.  37 


Let     z  =  siuo; ;  theu     dz  =  cosx.dx, 

COS".l'  =  1  —  z', 

7, —  =     :, 7,  =  i  log- ,  by  Art.  53, 

eos-x       J  I  —z-  \  —z 

/sec  x.dx  =  \_  log     "^^1"^  =  log  tan  (  ^  +  ^V 
1  —  sin  X  \4      2/ 

Examples. 

Prove  that'^r  1 )  I  esc  x.dx  =  A^  log  -^^^ ^  =  log  tan  ^• 

1  +cos.r  2 


—  ^cos^a; 

Vl  -  0.-^ 

Suggestion:  Let  x"  =  cos2. 

55.    The  formula  for  integration   by  jx(r/.s   (L  Art.  71)  [1]) 
becomes 


I  ;«Zt'  =  ?<i'  —  I  ■i;(T*<,  [IJ 


when  we  use  the  differential  notation.   It  is  used  as  in  I.  Chap.  V 
(a)   For  example,  let  us  find  J  a;"  log  x.dx. 

Let  ?<  =  logx,  and  dr  =  a;"da:; 

dx 
then  dw  =  — , 

X 

and  v  = -, 

n  +  1 

f^n+l                               /»     o<"                          T;""*"*     /,                             1 
X"  log  X.dx  = log  .f  —   I  — —  dx  = log  a; 
^              » + 1     "        J  7t  + 1           n  + 1  V   ^        n  + 

(h)   Required  |  .r siir'.r.rfx. 
Let  u  =  sin~'a;,         and   </*•  =  xdx  ; 
then  dw 


slx-^ 


38  INTEGRAL   CALCULUS.  [Art.  55. 

and  ^'  ~  V' 

I  X sin'^x.dx  =  ^  sin"' x  +  ^ (cos"' x  +  xV  1  —  ar) . 
(c)   Required   1 '■ — '—,. 


Let 

u  =  xe', 

and 

dv-       ^^-^       • 

{l+xy 

then 

du  =  {xe'  +  e')dx  =  e'(l  +  x)c/a; 

and 

^-TT-. 

r  xe'dx 

_  ^>-"^  ,  r...7..       ■'•'-'  , 

ICx.VMPLES. 


j'y/l-Sx-x'  Vl3 

(2)     (  xtan'^x.dx  =  1±jI  tan-'x  -  ^x. 

r    xdx      ^ 1_+ 1 

^   ^  J  {l-xf  1  -X      2(1 -x)- 

( , )     (•    _^^_  =  -  ^'27,:,— 7-  +  a  vers-'ii\ 

(•'»)     I  V2 ax  —  7?.  dx  =  '——  V">o^?^^  +  —  «in"'  '^  ~  "'.        iV 
iSuggestiun  :  Throw  2  (J.c  —  .Jr  into  tlio  funn  t(^  —  (.r  —  ct)^,      ,        ,'X' ,    "^ 

( G )     rL±iI!!ii"  ,/a;  =  log  (X  -f-  sin  x) . 


Chap.  III.]    GENERAL  METHODS   OF   INTEGKATLNG.  39 


(0  ;      •     f?.r  =  .rtaii-. 


.':>' 


I  -t  6^^^  ^ 


Suggestion :  Introduce  '-  in  [)lace  of  a;. 


z  =  sin  '.r. 


I)     f      ^^-       = \ 

J  .rClog.r)"  {n-  I)  (log.r)"-i 

t)  Jl2glM:£)  ,?.,  =  log.,  [log(loga-)  - 1] 

10)  I  ".  •  •  =  ^tan,r  +  logcosg.  where 

^J  (1-.^')^  ^     ^     ° 

.  2%  r  siDa;<?.r    _  _  log  (a  4- &  cos  a;)     ^ 
Va  +  6cos.T~  b  '     ''^"^ 

V  1  -  o;'^  .6        \l-  xy 
,..  r      x^dx  1  ,      /.r^-3' 

^^)  r .   .  t..  •  ■>  = 


L^ri/// 


—  tan  '( -tan.T 
ah 


'Ctf^ 


/&  ^^J^67Sf(^ 


J 


40  LNTEGKAL    LALCLLUS.  [AnT.  oG. 


CHAPTER    IV. 

RATIONAL    FRACTIONS. 

56.  We  shall  now  attemi)t  to  consider  s^-stematically  the 
methods  of  integrating  various  functions ;  and  to  this  end  we 
shall  begin  with  rational  algebraic  expressions.  Any  rational 
algebraic  j)ohjnomial  can  be  integrated  immediately  b}'  the  aid  of 
the  formula 


.P 


n  +  \ 

Take  next  a  ration(d  fraction,  tliat  is,  a  fraction  wliose  nu- 
merator and  denominator  are  rational  algebraic  polynomials. 
A  rational  fraction  is  j^roj^er  if  its  numerator  is  of  lower  degree 
than  its  denominator ;  improper  if  the  degree  of  the  numerator 
is  equal  to  or  greater  than  the  degree  of  the  denominator.  Sinre 
an  improper  fraction  can  always  l)e  reduced  to  a  pol3'nomial» 
plus  a  proi)er  fraction,  by  actually  dividing  the  mnnerator  by  the 
denominator,  wc  need  only  consider  the  treatment  of  proi)er 
fractions. 

57.  Every  projjer  rational  fraction  can  be  reduced  to  the  sum 
of  a  set  of  simpler  fractions  each  of  tvhich  has  a  constant  for  a 
numerator  and  some  poicer  of  a  binomial  for  its  denominator ; 

that  is,  a  set  of  fractions  any  one  of  which  is  of  the  form  — 

(.r  — a)"* 

fx 

Let  our  given  fniction  l>c  ^~. 

Fx 

Ifo,  b,  0,  &c..  arc  the  roots  of  tin-  e(juation, 

Fx  =  0,  ( 1 ) 

wc  have,  from  the  Tlicorv  of  Equations, 

Fx  =  A  {X  -  a)  (X  -  b)  {X  -  c) (2) 


Chap.  IV.]  RATIOXAL   FRACTIONS.  41 

The  equation  (1)  may  have  some  equal  roots,  and  then  some  of 
the  factors  in  (2)  will  be  repeated.  Suppose  a  occurs  p  times 
as  a  root  of  (1),  b  occurs  q  times,  c  occurs  r  times,  &c., 

(3) 


then 
Call 
then 

and 

Fx  =  A{x-  o)"  (.r  -  by  {x  -  c)- 

A{x-by{x-cy =  4>x', 

Fx=  {x  —  ay<l)X, 

fa                 fa 
fx_        fx            •^■'-'-^^^■'          '^^^^ 

Fx      {x  —  ay^tx      {x-ay<jix      {x  —  ay  <f,x 
fa                   fa 

fx- 

(x  —  ay       {x  —  ay(l>x 
fa  , 

(x  —  a)P<l)X  '■     '■ 

to  a  simpler  form  b}'  dividing  numerator  and  denominator  by 

X  —  «,  which  is  an  exact  divisor  of  the  numerator  because  ((  is  a 

root  of  the  equation 

fx-^cf>x=0. 
<f>a 

If  we  represent  by  f^x  the  quotient  arising  from  tlie  divi.sion 

of  fx  —  '—  d}X  by  X  —  a,  we  shall  have 
4>a 

fa_ 

fx  _       (f)a  fxX 


where — — is  a  proper  fraction,   and  may  be  treated 


Fx       {x  —  ny       {x  —  a)''-'  ^x- 
f\x 
{x  —  ay-'^  (fiX 
precisely  as  we  have  treated  the  original  fraction. 

Hence      ^1^^ =  ^±1—.  + ^%^— 

{x  —  ay-^4>x      {x  —  ay-^      {x-ay-^<f>x 

By  continuing  this  process  we  shall  get 

.fa  /a  f^  fp.\a 

fx  ^       <}>a  <i>a  4>a  <i>"      ^  fp-f 

Fx~~  {x-ay^ {x-ny-'^     {x-(iy--     .»•-"      <^' 


42  INTEGRAL  CALCULUS.  [Art.  58. 

f  X 

In  the  same  wav  -^^^  can  be  broken  up  into  a  set  of  fractions 
*"    <^x 

having  {x  —  hY,  (x  —  b)^~^,  &c.,  for  denominators,  phis  a  frac- 
tion which  can  be   broken  up  into  fractions  having   (x—c)", 

(x  —  cy-^ ,  &c.,  for  denominators;    and   we   shall  have,  in 

the  end, 

fx_       A,        ,  A,     __^ +  _A_4..     ^» 


Fx      {x-uy      {x-ay-^  x-a      {x-by 


'  (^x-by-'  '  x-b 

where  K  is  the  quotient  obtained  when  we  divide  out  the  last 
factor  of  the  denominator,  and  is  consequently  a  constant.  More 
than  this,  K  must  be  zero,  for  as  (1)  is  identically  true,  it^nust 

fv 
be  true  when  x=  x  ;   but  when  a;=  x,  :i!_   becomes  zero,  be- 

Fx 
cause  its  denominator  is  of  higher  degree  than  its  numerator, 
and  each  of  the  fractions  in  the  second  member  also  becomes 
zero;  whence  K=0. 

58.    Since  we  now  know  the  form  into  which  any  given  rational 
fraction  can  be  thrown,  we  can  determine  the  numerators  by  the 

aid  of  known  properties  of  an  identical  equation. 

g^. J 

Let  it  be  reciuircd  to  l)reak  up ~j- — — —  into  .simpler 

fractions.  ^  J   \    -r    j 


By  Art. 


3x-l  A  li      ,     O 


{x-iy{x-{-\)    {x-iy    x-i  '  x+v 

and  we  wish  to  determine  ^1,  7?,  and  C.     CU-aring  of  fractions, 
we  have 

Sx-\  =  A{x  +  \)  +  B{x-\)(x  +  ])-\-C(x-\y.       (1) 

As  this  equation   is  identically  true,  the  coefficients  of  like 
powers  of  x  in  the  two  members  nuist  be  equal ;  and  we  have 
Ii-\-C=0, 
A-2C=S, 
A-B  +  C=-l; 


CiiAP.  IV.]  KATIONAL   FRACTIONS.  43 

whence  we  find 


^=1, 

B=l, 

C=- 

1 

1 

A  3a!-l  1,1  1 

-<»         (^-,)H.  +  i)  =  (^i7  +  .7^  -  7TT-         <-' 

The  labor  of  detennining  the  required  constants  can  often  l>e 
lessened  b}'  simple  algebraic  devices. 

For  example  ;  since  the  identical  equation  we  start  with  is 
true  for  all  values  of  x,  we  have  a  right  to  substitute  for  x  values 
that  will  make  terms  of  the  equation  disappear.  Take  equa- 
tion [1]  : 

3x  -  1  =  ..^x-  +  1)  +  B{x  +  1)  (a-  -1)  +  C(.v  -1)-.       [1] 

Letx=l,  2  =  2  A, 

A=l, 

then  2x-2=:B  (x  +  1)  (x  -  \)-\-C {x -1)- ; 

divide  by  .T  - 1 ,  2  =  B  {x -\-l)  +  C  {x- 1). 

Leta;=l,  2  =  2B, 

B=l, 

then  —x+l=C{x-l), 

C=-l. 

Examples. 

(1)  Show  that  when  we  equate  the  coefficients  of  the  same 
powers  of  x  on  the  two  sides  of  our  identical  e(iuation,  we  shall 
alwa^'s  have  equations  enough  to  determine  all  our  refiuircd 
numerators. 

(2)  Break  up  9a^  +  9a;-128  .^^^.^  shnpU.,-  fractions. 

ix-3r{x  +  l) 

59.  The  partial  fractions  corresponding  to  :uiy  giMn  factor 
of  the  denominator  can  be  determineil  ilirectly. 


44  INTEGRAL    CALCULUS.  [Aur.  50. 

Lot  US  suppose  that  the  factor  in  question  is  of  tlie  first  degree 
and  occurs  but  once  ;  represent  it  b}'  x  —  a. 

^=-^  +  -f^  (I) 

Fx      x-a^  <f>x'  ^  ^ 

by  Art.  57,  where 

^« 

<^x  = , 

x  —  a 
so  that  Fx  =  (x  —  a)  (jyx. 

Clear  (1)  of  fractions. 

fx  =  A^x  +  {x-a)f,x.  (2) 

As  (1)  is  an  identical  equation,  (l')  will  be  true  for  an}'  value 
of  x.     Let  a;  =  a, 

fa  =  A4>a, 

A  =  ^,  (3) 

a  result  agreeing  with  Art.  57. 

Hence,  to  find  the  mmierator  of  the  fraction  corre-fpoxdivrj  to 
a  factor  (x  —  a)  of  the  first  degree,  tee  have  merely  to  strike  out 
from  the  denominator  of  our  original  fraction  the  factor  in  ques- 
tion, and  then  substitute  a  for  x  in  the  result. 

If  the  factor  of  the  denominator  is  of  the  ni\\  degree,  there  are 
V  i)artial  fractions  corresponding  to  it.  Let  (x  —  a)"  be  the 
factor  in  question. 

fx^      A,  A.  I         ^3  ,  .      A        fx   ,,s 

Fx     (x-a)''"^(x-a)"-'"^(.r-rt)"-2'^        "^x-ci^^x^   ' 

where  Fx  =  {x—a)"<j>x. 

fx 
Multiply  (1)  by  (.-c  — a)",  and  represent  (x  — a)"^  by  *x. 

<l>x  =.lj  +  .l,(.c  -  a)-\-A.,{x  -  a)-  -f- +  A„{x  -  ay-'' 

<l>x 


CfiaI'.  IV. j  RATIONAL    FRACTIONS.  4j 

DifiVreiitiate  successively  both  ineinlHTs  of  this  i(U'iitit\.  nn<l  put 
x=  a  after  ditlereiitiation,  and  we  get 

Ao  =  *'  ti, 

As  =  —<i>"a, 
2  ! 

-(44  =  — <l>"'a, 
'3! 


(u-1) 


Although  these  results  form  a  eoinplete  solution  of  the  prob- 
lem, and  one  exceedingly  neat  in  theor}",  the  labor  of  getting 
the  successive  derivatives  of  *a;  is  so  great  that  it  is  usually 
easier  in  practice  to  use  the  methods  of  Art.  58  when  we  have  to 
deal  with  factors  of  higher  degree  than  the  first.  So  far  as  the 
fractions  corresponding  to  factors  of  the  first  degree  and  to  the 
highest  powers  of  factors  not  of  the  first  degree  are  concerned, 
the  method  of  this  article  can  be  profitably  combined  with  that 
of  Art.  58. 

60.  As  an  example  where  the  method  of  tlie  last  article 
applies  well,  consider 

S-t;-!  ^A_^     B      ,      C 


X  (.^•  —  2)  (.r  + 1 )       X       x—'2      x-  + 1 

l_r         '^'"^'-^         1     =- 
[(.r-2)(.r+l)i  =  o     2' 

[_x{x-\-\)i-_,      6' 

Lx(.x--2)i..,  ;3 

3.r-l  1    1    .  •■'       1  I       1 


x{x  -  2)  (.c  -I-  1 )       2   a;      G   x  -  2      3   x  +  1 


46  INTEGRAL   CALCULUS.  [Art.  6L 

61.  Although  the  theory  expounded  iu  the  preceding 
articles  is  complete  and  can  be  applied  without  serious  diffi- 
culty to  the  case  where  some  or  all  of  the  roots  of  F{x)  =  0 
[Art.  57,  (1)]  are  imaginary,  there  is  a  practical  convenience 
in  modifying  the  method  so  as  to  avoid  the  explicit  intro- 
duction of  imaginaries  into  the  process  of  integrating  a 
rational  fraction. 

We  know  (Art.  28,  Ex.  2)  that  if  the  denominator  of  our 
given  fraction  contains  an  imaginary  factor  (x—a  —  &V— 1)" 
it  will  also  contain  the  conjugate  of  that  factor,  namely, 
(x  —  a  +  b  V—  1)",  and  will  therefore  contain  their  product 
l(x  —  ay-\-b^Y.  Moreover,  since  by  Art.  59  the  numerator  of 
the  partial  fraction  whose  denominator  is  (x  —  a-\-b_^ — 1)^ 
is  the  same  rational  algebraic  function  oi  a—b  V—  1  that 
the  numerator  of  the  partial  fraction  whose  denominator  is 
(x  —  a  —  b-sJ—lY  is  of  a  +  i  V— 1,  these  two  numerators 
must  be  conjugate  imaginaries  by  Art.  28,  Ex.  3.     Hence,  for 

every  partial  fraction  of  the  form 1= —  we  shall 

{x  —  a  —  b'sl—l)P 

have  a  second  of  the  form 


{x-a  +  b  yj-  ly 

Let  {x-a-b V^)"  =  X-\-  Y V^, 

X  and  Y  being  real  functions  of  x ;  then 

{x-n  +  b  V=^)P  =X-Y  V=a. 

The  sum  of  the  two  fractions 
A-\-B  V^  A-B  V^ 


{x  —  a-b^—\y       {x  —  a+byj—iy 

A -{- B  yj^^       A  -  B -sl^^  _    2AX-\-2BY 


AT-f  y  V- 1    A'-  y  V- 1    [(x-  ay  +  b^y 

and  is  a  real  proper  fraction.     Hence, 


Chap.  IV.]  RATIONAL    FRACTIONS.  47 

A /i«  I  f-pr 


every  numerator  being  of  lower  degree  than  its  denominator. 

If  we  take  — '  ' .,  ,   ,„-,„  and  divide  numerator  and  de- 

nominator  by   (.r  —  af  +  h"^  we    shall  get  a  fraction  of  the 

form  ^- ^    ^.,  ,   ,..-, — r  and  R  will  be  of  the  first  degree  and 

[(a:  — a)-  +  ^^]"-i  * 

therefore  of  the  form  L^x  +  -1^>  and  we  shall  have 


By  successive  repetitions   of  this   process  we  can  reduce 


to 


[(x-af-\-b'y'^[(x-ay-\-b'Y-'^""^    (x-af-^fj'   ' 

Treating  all  the  partial  fractions  in  (1)  in  this  way  and 
adding  the  results,  we  shall  at  last  reduce  (1)  to  the  form 

fie A,x-\-B,  A^x  +  B^ 

[(x-ay  +  Py'4>x~  [(x-af  +  O'f^  [(x-ay-^b']"-' 

and  our  partial  fractions  are  simple  in  form  and  do  not  involve 
imaginaries. 

The  coefficients  in  (2)  can  be  found  by  either  of  the  proc- 
esses illustrated  in  Art.  58. 

62.    Let  us  now  consider  a  ratlier  difficult  example,  where 
it  is  worth  while  to  combine  all  our  methods. 


48 

INTEGRAL    CALCULUS. 

.r^+l 

,.  _  1  w  r"  4-  1  \i 

[Art.  G2. 


a;8-f-]  =(j.4-i)(j.2-^-|-l)  and  x''  —  x-\-l=0  has  imagi- 
nary roots. 

x^  +  1 x^-\-l 

(x-i)(x»+iy~(x-i)(x-\-iy(x'-x-\-if 

-  '^    I     ^^     \    ^'    \    <^i^  +  A    I  ^2^-4- A     ,.. 

~  x-l^  {x+\r-^  .r  +  \^  {x-'-x+iy^  x'-j:  +  1        ^  ^ 

^^=  U-l)(.:^'-x+l)d.=  -r~*" 

Substitute  in  (1)  the  values  just  obtained,  clear  of  fractions 
and  reduce  and  we  have 

-  9  .r«  +  2  .r«  —  6  X*  —  8  x'^  +  8  x2  +  6  a-  +  7 

Divide  through  by  x^ —  1,  and  we  get 

—  9a;*  +  2x''-15x2-6x-7 

=  ]8^A(a:^-.r  +  l)^+(.r  +  l)[C,a-  +  A 
+  (C,..T  +  A)(^'-•i•  +  l)J^ 

Let  a;  =  —  1,  and  we  find 

Ji,  =  -h 

Substitute  this  value  for  A  and  reduce ; 

—  6  J-*  —  4  a-"'  —  G  a-^  —  12  a-  —  4 

=  18  (.r  +  1)  [  C.r  4-  />;  +  (  C\x  +  A)  (.r^  -  .r  +  1) ]. 
Divide  by  jr  +  1  and  expand  and  we  get 
[18  C^  +  6]  a"'  -  [18  (Ca  -  A)  +  2]  .r' 

+  [18(r,  -  Ih  +  r,)  +  8]  .<•  +  ISCA  +  A)  +  4  =  0. 


CiiA!-.  IV.]  RATIONAL   FRACTIONS.  49 

This  equation   must  hold  good  whatever  the  value  of  x, 

whence 

18  C2   +6  =  0, 
18(C2-A)  +  2  =  0, 

18(0,- D,-\-C\)  + 8=0, 

18(A+A)+'i  =  0, 
and 


c. 

=  — 

h 

A 

=  — 

h 

c\ 

=  — 

h 

D, 

=  0 

Hence, 

x'+l 

1 

1 

1 

1 

{x  —  \){x'-\-iy      2   a;-l       9    (.r  +  1)-      6   .r  +  1 

_1  X 1       3  ■>■  +  2 

3  "  (a;^  —  X  +  1)^      9  '  x"-^  —  x  +  1 ' 


(2) 


63.    Having  shown  that  any  rational  fraction  can  be  reduced 
to  a  sum  of  fractions  which  always  come  under  the  four  forms 
A  A  Ax-\-B  Ax  +  B 

(x  — a)"'  x  —  a     (x  —  af-\-lr'   [(x  —  «)-^  + Z--]"' 

it  remains  to  show  that  these  forms  can  be  integrated. 

To  find       ; -' 

J  (x  —  ay 

let  z  =  x  —  a, 

then  dz  =  dx, 

r_Adj-_  _      Cd±  _  _       1  A 

J  {x  -  ay  ~  "  J  z"  ~      (n  -  1)  '  z'^-' 


{n  —  V)   (x  —  ity 

To  find   C-^^, 
J  X  —  a 

let  z  =  x  —  a, 

then  dz  =  dx, 


cn 


60  INTEGRAL   CALCULUS.  [Akt.  63. 

and  J^^  =  Af~=A\ogz  =  A\og{x-a).  [2] 

Turning  back  to  Art.  58  (2),  we  find 

/(8x  —  l)dx     _  r     clx  r  dx     _  C_dx__  _  _     1 

ix-lY{x^l)~J  (x-iy'^J  x-1     J  x  +  l~      x  —  1 

-\-\os{x-l)-log{x  +  l)=--^  +  log''~' 


x-1   '      °x-\-l 
Turning  to  Art.  60  (1),  we  have 

/(3x  —  l)dx      _     rdx  r  dx    _     r  dx 

x(cc-2)(x  +  l)~  V    x^Vx-2      Va:  +  1 

=  i  logx  +  I  log(a:  -  2)  -  I  log(.r  +  1). 

^    ,    ,   r(Ax-{-B)dx 

To  find  )  f \2\_j2- 

J  (x  —  ay  +  ¥ 

Ax  +  B      _    A(x  —  a)  Aa-{-  B 

(x  -  ay -\-b^~  (x-  ay  -\-b'^  (x-  ay  +  b^' 

If  we  let  z  =  (x  —  ay  +  b^,  dz  =  2  (x  —  a)  dx,  and 

/A(x  —  a)dx      A   rdz      A  A  no  i   /an 

If  we  let  z  =  x  —  a,  dz  =  dx,  and 
C(Aa-\-B)dx       ...    j,^  r_J^ 

Aa  -\-  B  ,        ,z      Aa  -\-  B  ,        ,x  —  a 
—  tan~  ^  7  = ; tan~ 


b  b  b  b 


Hence,  r^<-^-+ fit 
'J  {x  —  aY-\-b^ 


A.      ^.           ,,   .    ,2n    I   -'^n -\r -B  ,        .x  —  a      _„_ 
=  -  log  [(.r  -  ay  +  b'^  + ,—  tan- '  — —  •    [3] 


_     .    ,    C   (Ax+B)dx 
To  find  I  p7 -rj-n 

J  [(x  —  ay  +  b^ 


[(•r-ay  +  b^y 

Ax-\-  B         _        A{x-a)  Aa  +  7? 

l{x  -  ay  +  l^'Y  ~  l{x  -  ay  +  />2]»  "•"[(.*•-  ay  +  i^]" ' 


CiiAP.  IV.]  RATIONAL    FRACTIONS.  61 

If  we  let  z  =  (x  —  ay  +  b-,  ch  =  2(x  —  a)  dx,  and 

/A(x  —  a)  clx     _  A   r(f-_  A  _„^, 

l(x-ay-{-lj^f~  2J   ^■'~      2{ri-lf 


2(?i  — 1)  \_{x  —  ay-\-b^J-^ 

If  we  let  z^x  —  a,  dz-=^ dx,  and 

C  {Aa-\-B)dx  C       dz 

I  '^        can  be  made  to  depend  upon  j  /  2_i_/2xn-i  ^J  *^® 

aid  of  the  reduction  formula  [6],  Art,  64,  which  for  this  special 
form  reduces  to 

/dz 

2  (/i  -  1)  J?  if  +  h'y-^  "^  2  (n  -  1)^*2  J  (^2  +  Z,2y.-i      L  J 
Hence    f_(:l^±^H£_= A 1 

/dx  1  a —  g 

[(X  -  ay  +  ^'•^]"  ~  2  («  - 1)  b' '  \_{x  -  ay  4-  b^r-' 

■        2?l-3        r  r/.r 

"^2(«-l)^'-J  [(x-«)^  +  Z''^]''-'    L  J 

/*  dx 

A  repeated  use  of  [6]   will  reduce  J  ^     _    ,,,  ^g-.,  to 

/dx 
_    x2  I  7,2'  which  has  already  been  found 


1.         U         1     4.  -1   ^~« 

to  be  7  tan   '  — ; — 

0  0 


62  INTEGRAL   CALCULUS.  [Art.  63. 

Turning  back  to  Art.  (52  (2),  we  find  that 

r       (.r-+l)^/.r        ^  ,     rj.r_  _  ^     f      dx        _       f    dx_ 
J  (X  -  1)  (X-' +  1/        U.r-1        '''J(x  +  iy       *Jx  +  l 
r        .r  d.r         _  J   r(3x-i-  2)  dx 

^  J  (.r'  -  X  +  If        O     x'-X-\-l 
=  ^log(.r-l)+  l--^-h^og{x  +  l) 

x  —  2  ,-  ^        ,2a;  —  1 

-ilog(.r^-x  +  l)-,7^V3tan-i^^ 

^     ^   x'  + 1     '   ^  a-3  + 1       ^  V3 


Examples. 

r_^^3^3_^,^_  .>::_2. 

J  x-^  —  4  X  +  2 

^  J  X-  +  1 

,.     r    dx         ,,       (.«-!)-  1    ,     _,2.-r  +  l 

4)  I  =X\oii^-^ tun  ' ■ — 

^  Jur'-l       ^     ^.tr^  +  x-  +  l        V.3  V3 

5) =  — -  tan  '  -  +  -T-T  log 

J  iC  —  x*       2(r  u       4«'*        a—x 

^  J  (a^'4-l)(.r-'  +  x  +  l)       -     ^     .r'  +  l         V3  V3 

7)     r_^^f_  =  ilog^i^+^tan-'-^. 
'^  .1  j,4  4.-ar'_2      ^    ^a'  +  l        3  V2 


ClIAl-.    IV.] 


RATIONAL    FRACTIONS. 


53 


^        J  x*  +  x-  +  l  ^     ^X"+X  +  \ 

^   ^  J  {x-\y(.r-  +  \r  4(.r-])       -     -^ 

+  i  taii-^i-  -        ,}  +  i log(.r=  +  1 ) . 

4(.r-  +  l) 

(10)  ri:!^^_j_iog^-W2  +  i 

. '  .1-^  + 1      4  V2        jc2  _|.  ^  7-2  +  1 

4-  — ^  [tan-'  (x  V2  + 1 )  +  tan"'  (x  V2  - 1 )  ] . 
2  V2 


(11 


^  J  :r^  +  1      4 


a;V2^\ 

V"2  '""  ar  _  ic  V2  +  1   '  2  V2  "*"     Vl  -  ^J' 


dx  1     ,      .T2-}-a;V2  +  l   ,      1 


log 


tan" 


64  INTEGRAL  CALCULUS.  [Art.  64. 


CHAPTER    V. 

REDUCTION    FORMULAS. 

64.  The  method  given  in  the  last  chapter  for  the  integi-ation 
of  rational  fractions  is  open  to  the  practical  objection  that  it  is 
often  exceedingly  laborious.  In  many  cases  much  of  the  labor 
can  be  saved  by  making  the  required  integration  depend  upon 
the  integration  of  a  simpler  form.  This  is  usually  done  by  the 
aid  of  what  is  called  a  reduction  formula. 

Let  the  function  to  be  integrated  be  of  the  form  a;"'"^(a+6x")p, 
where  m,  n,  and  p  may  be  positive  or  negative.  If  they  are  in- 
tegers, the  function  in  question  is  either  an  algebraic  polynomial 
or  a  rational  fraction;  if  they  are  fractions,  the  expression  is 
irrational.  The  formulas  we  shall  obtain  will  appl}-  to  either 
case. 

Denote  a  +  6.r"  by  z  ;  then  we  want  |  x^-'z^da;. 
Let  '      z^=u 

and  K""'  dx  =  dv,  and  integrate  by  p>arts. 

du  =  lyz^-^  dz  =  bnpx"-^  z^"'  dx, 
ar 
m 

J  m  m  J 

This  formula  makes  our  integral  deix'nd  upon  tlie  integral  of 
an  expression  like  the  given  one,  except  that  the  exponent  of  a; 
has  been  increased  while  tliat  of  z  has  been  decreased. 

We  get  from  [1],  by  transposition, 

fx-  +  "- '  ^r-i  rfj;  =  i^''  _  JL  A.™- 1 2P  ax. 
J  bnp      blip.' 


Cn.vp.  v.]  REDUCTION   FORMULAS.  56 

Change  7?i  +  n  into  vi  and  p  —  1  into  j),  whence  m  is  changed 
into  m  —  n  and 2^  into })  -\-l,  and  we  get 

J'  m-i  D  J  a;"'""^^"*"^  m  —  u       C  „,  „  ,  „,,  ,  _  _ 

bn{p+\)       hn{p  +  \)J  L-J 

a  formula  that  lowers  the  exponent  of  x  while  it  raises  that  of  >. 

Since  z  =  a  +  tx", 

2^  =  2P-i(rf +  6.C''), 
hence 

therefore,  by  [1], 

in         m  J  J  J 

Cx-^z^-^dx  =  ^-  Hl^i±J!Pl  Cx-—^z^-hlx. 
J  am  am       J 

Change  p  into  p  +1. 

C:c-^,.dx  =  ^^:i^  -  ^(^^  +  ^>P  +  -)  fx-^-^z^dx.      [3] 
J  am  «?'i  "^ 

Change  m  into  m  —  n,  and  transpose. 

f^..-. ,.,?.,  =    x^-'z"-^    _  «(m-n)    p.-„-,,.,,,.       ^43 

We  have  seen  that 

Cx"'-'^z''dx  =  a  Cx'^-^zP-^dx  +  &  Cx"'  +  "-^z'-'^dx, 
and,  from  [1], 

J  np        npJ 


56  INTEGRAL   CALCULUS.  [AUT.  G5. 

hence 

Cx''-^z''dx  =  a  ( x"'-^ z''-^ dx  +  ^^  -  —  Cx'^-^z''dx, 
J  J  np        njiJ 

fx'"-h>'dx=     •^'"'~''    +    "^'^'      i'x"'-'zP-'dx.  [;■)] 

Change  p  into  ;>  +  1.  and  transpose. 

an{p  +  1)        a»(j)4-l)  -/ 

Formula  [3]  enables  us  to  raise,  and  Ibrniula  [4]  to  lower,  the 
exponent  of  x  by  n  without  affecting  the  exponent  of  z ;  while 
formula  [o]  enables  us  to  lower,  and  formula  [G]  to  raise,  the 
exponent  of  z  by  unity  without  affecting  the  exponent  of  x. 

Formulas  [1]  and  [3]  cannot  be  used  when  ?/i  =  0  ; 

formulas    [2]  and  [6]  cannot  be  used  when  p=  —\\ 

formulas    [4]  and  [5]  cannot  be  used  when  m  —  —np  ; 
for  in  all  these  cases  infinite  values  will  ho  brought  into  the  sec- 
ond member  of  the  formula. 

G.'>.    If //  =  1,  z  =  u  -{-hx, 

and  our  last  four  reduction  formulas  become 

I  .X-"- '  z^dx  = 5^ —    ^         '  I  xTzHlx.  \:\ 

J  am  am  J  •-  -^ 

J  a(p+\)       a{p  +  \)  J  •-   J 

If  »j  and  J)  are  integers,  and  m>0  and  ^>>0,  a  repeated  use 
of  [.")]  will  reduce  p  to  zero,  and  we  shall  have  to  find  merely 

the   Tr-'f/x. 


Chap.  V.]  REDUCTION    FORMULAS.  57 

If  //(<0  and  2)>0,  [o]  will  eiuible  us  to  raise  in  to  0,  and 
tluii   [5]   will  enable  us  to  lower  2>  to  0,  and  we  shall  need 


oiilv 


J 


'dx 


X 


ll'»i>0  and  7)<0,  [0]  will  raise  j)  to  —1,  and  [A]  will  then 
lower  m  to  1,  and  we  shall  need  I  — . 

If  /H<0  and2><0,  [G]  will  raise  7;  to  —1,  and  [3j 


wui  raise 


//(  to  0,  and  we  shall  need   |  — • 
J  xz 


af'-'  dx 


f 

J   z      J  a  -\-  bx      b 

rcjx^r     dx     ^_ii^i± 

J  xz     Jx{a+bx)  a     "      x 


Hence,  when  7*  =  1,  and  ?h  and  j>  are  integers,  our  reduction  for- 
mulas always  lead  to  the  desired  result. 

Examples. 

J  x\<(+bx)~      a'^^     X  (I'x      ^ir'x"      Sirx''     A  ax* 

(2)  Consider  the  case  where  u  =  2,  rewriting  the  reduction 
formulas  to  suit  the  case,  and  giving  an  exhaustive  investi- 
gation. 

^   ^  J  {a  +  bx-y  ^  ~  Abia+bx")-       8nb{a-\-bxr) 

_^_L_tan->.rJ^. 
H{ab)l  \<i 


58  INTEGRAL  CALCULUS.         [Art.  66. 


CHAPTER   VI. 

IRRATIONAL     FORMS. 

66.  We  have  seen  that  algebraic  polynomials  and  rational 
fractions  can  always  be  integrated.  When  we  come  to  irrational 
expressions,  however,  very  few  forms  are  integrable,  and  most 
of  these  have  to  be  rationalized  by  ingenious  substitutions. 

If  an  algebraic  function  is  irrational  because  of  the  presence 
of  an  expression  of  the  first  degree  under  the  radical  sign,  it  can 
be  easily  made  rational. 

Let  /(x,  \'a  -\-  bx)  be  the  function  in  question. 
Let  z  =  \/  a  +  bx ; 

then  2:"  =  a  +  bx. 

nz^'^lz  =  b(lx\ 

hlz 


dx  = 


b 
2"  —  a 


b 
Hence        Cf{x,  ^T+bx)dx  =  ''  CfK^,  ^z^-'dz, 
which  is  rational  and  can  be  treated  by  the  methods  of  Chapter  IV. 


Examples. 


(1)   r^A  +  lr?a;==x  +  4  V-^  +  41og(Va^-l). 

./   y X'  —  1 

(2)  rv(„x+M-.>x="v<"+'->;". 

(3)  J[.tV(-^-  +  ")+V(-»-  +  ")]''x 

2n  +  \  n-\-\  ^^VV    -r    ; 


Chap.  VI.]  IRRATIONAL   FORMS.  69 

67.    A  case  not  unlike  the  last  is   (/(.(•,  Vc  +  Vu  +bx)dx. 


Let  z=  -Vc-h  "Va  +  bx  ; 


z"  =  c  +  \  a  -j-  bx, 
(z"— c)"'  =  a  -hbx, 

b 


Hence 


b 
Cf{x.  Vf4-  y/aTTx)dx 


(1)  Find   C 

(2)  Find   f- 


Examples. 

xdx 


Vc  -f  Va  +  bx 
dx 


VI  +Vl  -.r 

68.  If  the  expression  under  the  radical  is  of  a  higher  degree 
than  the  first  the  function  cannot  in  general  be  rationalized. 
The  most  important  exceptional  case  is  where  the  function  to  be 
integrated  is  irrational  by  reason  of  containing  the  square  nwt 
of  a  quantity  of  the  second  degree. 

Required     | /(a-,  Vo  -\-bx-\-  rx^)dx. 

First  Method.    Let  c  be  positive  ;  take  out  Vc  as  a  factor,  and 
the  radical  may  be  written  V^l  -\-  Bx  -\-  xr. 
Let  VA  +  Bx  +  .-i^  =  x  +  z, 

A  +  Bx-hx-  =  x-  +  2  xz  +  z^, 

B-2z 


dx  ■■ 

{B-2zy 


60  INTEGRAL   CALCULUS.  [Art.  68. 

•>{z--  nz  +  A)dz 

-2zy     ' 

and  the  substitution  of  these  vahies  will  render  the  given  func- 
tion rational. 

Second  Method.     Let  c  bo  positive  ;  take  out  Vc  as  a  factor, 
and,  as  before,  the  radical  may  be  written  V-4  +  Bx-{-  x^. 


Let  ^/A+Bx-\-  x^=y/A  +  xz; 

A  +Bx  +  :>?  =  A  +  2^A.xz  +  x'z'', 
2  J  A  .z-B 

X  =  —^ 7, ' 

1  -z- 
^  2i^A.z^-Bz+^A)dz 
(l-z^y 

and  the  substitution  of  these  values  will  render  the  given  func- 
tion rational. 

If  c   is    n(»£rative  the    radical    can    be    reduced   to   the   form 


V^-1 4-  Bx  — .»-'.    and    the    method    just   given    will    present   no 
difficulty. 

Third  Method.  Let  c  be  positive  ;  the  i-adical  will  reduce  to 
-s/A  +  Bx  +  a^.  Resolve  the  quantity  under  the  radical  into  the 
product  of  two  liinomial  factors  (x  —  a)  (x  —  ^) ,  a  and  fi  being 
the  roots  of  the  equation  A  +  Bx  -\-x^  =  0. 

Let  V(.T-a)(.r-/?)  =  (x-a)z  ; 

(X  -  a)  (X  -(3)  =  {X  -  a)-2-, 

{i-zr 

^(x-u)ix-(i)  =  (X-  -  ,.)z  =  1^5^, 


TiiAP.  VI. J  IRRATIONAL    FORMS.  61 

and  tlio  substitution  of  those  values  will  make  the  given  funetion 
rational. 


If  c  is  negative  the  radical  will  reduce  to  V .1  +  Bx  —  xr,  and 
maybe  written  y/ {a  —  x)  {x  — /S)  where  a  and  (3  are  the  roots 
of  ar  —  Bx  —  A  =  0,  and  the  metiiod  just  explained  will  applv. 

In  general,  that  one  of  the  three  methods  is  preferable  wliic-h 
will  avoid  Introducing  imaginary  constants  ;  the  first,  if  c  >  0 ; 
a  a 

the  second,  if  c  <  0  and >  0  ;  the  third,  if  c  <  0  and  —  <  0. 

—  c  —  0 

a 
If  the  roots  a  and  fi  are  imaginary,  and  A  =  ^;j-  is  negative,  it 

will    be    impossible   to   avoid    imaginaries,    for    in    that    case 
A  +  Bx  —  x^  will  be  negative  for  all  real  values  of  x. 

69.    Let  us  compare  the  working  of  the  three  methods  just 

given  b\-  applving  them  in  turn  to  the  example   j  '■ 

J  V  2  +  3 .1-  +  ar' 

1st.    Let  V2  +  3.v-}-ar  =  .r  +  2; 

r  dx  ^  r2{z--3z  +  2)dz        3-2z      ^  r  2dz 

J  V2  +  3a;  +  .x-^~-'         (3-2^)^'        ';^-3z  +  2      J  3-2z 

=  -log(3-22). 


/: 


dx 

=  log; 


log(3  4-  2 a;  -  2  V2  +  3  x  +  or) 

V2  +  3a;  +  .'" 

1 


+  2.r-2V2+3.x'  + 


_  3  +  2  X  +  2  V2  +  3  X  +  or 


=  log  [3  -f  2  X  +  2 V2  -f  3  .r  +  .r-] . 


2d.    Let  \/2-\-3x  +  jr  =  ^2+xz; 

r         dx  _     r(V2.2--3z+ V2)rf2  1  -z- 

JV2T3^+T-       J  (1-^r  '^2.^-:iz  +  ^2 

=  2r_^  =  logL±£.  (Art.;-i3) 

J  I  —  Z^  1—2 


62 

dx 

+  -dx 

+ 

INTEGRAL 

.r            a;  +  V^ 

CALCULUS. 

[Art. 

65 

J  V2" 

+  V2  +  ; 
-  V2  +  ; 

Sx-\-x' 

yl2  +  -6x 

+  ar^  +  2  +  3a 

2 

rr'+  2  ^-1  .X  +  2  -  2  -  'dx  -  :)!? 


=  log 


3  +  2.1-  +  2  V2  4-3a;  +  ^ 
2V2-3 


=  log(3  +  2x+2V2+3a;+ar^)  -  log(2  V2-3) 
or,  dropping  the  constant  log(2^2  —  3), 

r       dx 

'^  V2  -f-  3  X  +  X' 


log(3  -f-  2  X  +  2  V2  +  3 .1-  +  or') , 


(2) 


3d.    Let  V2  +  3a;  +  or'  =  V(a;+  1)  {x  +  2)  =  {x  +  1)2; 

f—^==2f-^^^'—^  =  2C^^=log'-±- 
JV2H-3X  +  .X-         J  (1-22)2     _2  J\-z'  1- 


/: 


dx 


V2  +  3a;  +  ar' 


log 


X  +  2 


\  .r  +  1 


=  log' 


+  1  +  2  V2  +  3a;  -f  af^  +  .-K  +  2 


a;  +  1  —  .T  -  2 
=  log  (3  +  2x  +  2  V2  +  3  .r  +  ar)  +  log  (  -  1 ) , 
or,  dropping  the  imaginary  constant  log  ( —  1 ) , 

f-^^^ --:=  log(3  +  2.r  +  2V2  +  3a;  +  ar').  (3) 

^  V2  +  3a;-f-  a;- 

EXAMPLES. 


O)    r ^^L-^,=,  =  J-iog^''i+^iL=LV|^. 

^   ^  J  (2  +  3x)V4-x-     4V2        V4  4-2X-+ S/2-a; 

(2)  f-.^  =  log(A  +  a-  +  V?T^). 
./  Vor  +  x 

(3)  f  ,       ^ r  =  —\og(—  +  x^c-\-yfa  +  bx  +  cA. 


CiiAP.  VI.]  IRRATIONAL   FORMS.  63 

70.  If  the  function  is  irriitional  through  the  presence,  under 
tlie  radical  sign,  of  a  fraction  whose  numerator  and  denominator 
are  of  the  first  degi-ee,  it  can  always  be  rationalized. 


Required  j>(.,^^)... 


Let  ._;-i"-+& 


"  lax  -j- 


m 


^„  _  ax  +  b 

Ix  +  m' 

&  —  mz^ 

Iz"  —  a^ 
J    _  n(am  —  bl)z"-^dz 


X: 


{Iz'^-aY 

and  the  substitution  of  these  values  will  make  the  given  function 
rational. 

Example. 

f  dx     s/T^ = _  #  3 1/1 --^-y 
J  {\+xy\\+x      ^\\\+x)' 


71.  If  the  function  to  be  integrated  is  of  the  formrc"  "'(a+&x")'', 
m,  71,  and  p  being  any  numbers  positive  or  negative,  and  one  at 
least  of  them  being  fractional,  the  reduction  formulas  of  Art.  64 
will  often  lead  to  the  desired  integral. 

Examples. 


(1)    f     ^^^^-^      =§sin-'x-^^'-^(:{4-2.r^). 

(-2)  f-A'_^^iog^--^^^-^^ng. 

(3)    f       ^^^^        =-(2ax-.x-^)^(^^>^V3«'sin-\f^. 
^      .)  {2ax-x')\  ^  '  {-2^  2  J  \2u 


64  INTEGRAL  CALCULUS.  [Aur.  72. 

72.  We  have  said  that  when  an  irrational  function  contains  a 
quantity  of  a  higher  degree  than  the  second,  under  tiie  square-root 
sign,  it  cannot  ordiudrihi  Ije  integrated.  It  would  be  more  cor- 
rect to  say  that  its  integral  cannot  ordinaril}-  be  finitely  expressed 
in  terms  of  the  functions  with  whicli  we  are  familiar. 

The  integi-als  of  a  large  class  of  such  irrational  expressions 
have  been  specially  studied  under  the  name  of  Elliptic  Integrals. 
They  liave  |)eculiar  i)roperties,  and  can  l)e  expressed  in  terms  of 
ordinary  functions  only  l)y  the  aid  of  intinite  series. 


Chap.  VII.]  TRANSCENDENTAL   FUNCTIONS.  05 


CHAPTER    VTT. 

TRANSCENDENTAL    FUNCTIONS. 

73.    In  dealing  with  the  integration  of  transcendental  functions 
the  method  of  integration  b>/  parts  is  generalh"  the  most  etlective. 

For  example.     Required   |  x{logx)-clx. 

Let  u  =  {logxy, 

dv  =  x.dx ; 

"Iloax.dx 
dn  = ^ . 

X' 
XT 

fx(\ogxy-  =  -til^K^  -Cxlogx.dx  =  |[(log.r)'-  log.r  +  i]. 

Again.     Required    i  e^s'mx.dx. 
u  =  sina.% 
dv  =  e'  dx  ; 
du  =  co%x.dx^ 

V  =  f^, 

I  ^'sin  x.dx  =  f^^sin  x  —  |  ^''cosr.rfx, 

I  e'cos.T.'^/.r  =  r''cos.r  +  |  '■'sin  x.dx ; 

,                         r  r            7         <''(sin.r  —  cos.c) 
whence  I  ('"^sin  x.dx  =  -^ ^ 

and  I  e'  cos  .r .  dx  =  — ^ ^  • 


66  INTECJKAL   CALCULUS.  [Art.  74. 


_^     6l0g.T 


{m-t\y-     {m+\y 

^   \)    {\-xf       1  -X 

74.  The  method ^f' iiitegration  h}- parts  gfyes  us  important 
reduction  formulas  for  transcendental  functions.  Let  us  con- 
sider I  sin"x.dx. 

11  =  sin""^a;, 
dv  =  sinx'.cZa; ; 

dw  =  (?4  — 1  )sin""^T  costc.cZa;, 
v  =  —  cosa; ; 
j  sin"x.dx  =  —  sin""'.c  cos  a-  -f-  (?i  — 1 )  I  sin"--.r  cos^rc.do; 

=  — sin"''xcos;<;  +  (?i  —  1)  I  (sin^-'a;  — sin"a;)dr ; 

fsin"a*.fZa:  = sin""'.r  cosx  -\ — '^—  |  sin" "^x. da;.  [1] 

n  n    J 

Transposing,  and  changing  7i  into  n  +  2,  we  get 

rs\n"x.dx= sin"+*acosa;-|-^^-t-  |  sin"  +  ^x.fZx.       [2] 
71+1  ?i+lJ  -^ 

In  like  manner  we  get 

/ cos" a;.da;=  -sin a: cos""' a:  -\ ^^^—  |  cos"~^x'.da;,  [3] 

n  71    J 

/cos"a;.dx  = sina;cos"  +  'x +  ^?-t^  |  cos"  +  2x.da;.  [4] 
71+1  ?(  +  !.' 

If  n  is  a  positive  integer,  formulas  [1]  and  [;^]  will  enable  us 
to  reduce  the  ex[)onent  of  the  sine  or  cosine  tu  one  or  to  zero, 


(^llAl-.  VII.]         TRANSCENDENTAL   FTNCTIONS.  G7 

and  then  wo  can  integrate  by  inspection.  If  u  is  a  negative 
integer,  fornuilas  [2]  and  [4]  will  enable  us  to  raise  the  ex- 
ponent to  zero  or  to  minus  one.     In  the  latter  case  we  shall  iK-ed 

,  or   I  -r^—,  which  have  been  found  in  Art.  51  (<■). 

cos  a;         J  siua; 

Examples. 

/ 1 X     r  •  A      7  sin  X  cos  .r  /  .  ,      ,  3\   ,  3 

(1 )  I  snr x.clx  = (  snr x  +-\-}--x. 

/ ->\     r      r,      7        sinxcos'x/      o      ,  r)\   ,    5  ,  .  ,     , 

(2)  I  cos  x.clx  = (  cos-.r  +  -    H ;  (snio:  cosa;4-  .r). 

J  6         \  AJ      IG 

J  sur.^•  2  sin- a;  2 

(4)  Obtain  the  formulas 

/sinh''a'.f?.r=-siuh"  ^x'cosh.r  — ^^^^^  |  sinh"~'.i'.f/x. 
n  n    J 

/sinh".r.f?.i'= sinh"^^^^•coshx•—  "         j  sinlr'+-.i-.d.r, 
n  +  \  n-\-\J 

/eosh".)'.cZ.r= -sinhx'cosh"  'a;-| — ^^^  j  cosh"  -x.dx. 

/cosh."x.dx= sinh.7;cosh"'^^a;4-^          I  cosh"^- x.dx. 
?i  +  l  n-\-\J 

,..     C    dx  ,  cosh.1'       ,,      cosIki;— 1 

(^y)     I —      =  — -i- ilosj 

J  suih'a;  sinlr^i;  "cosh.i-+l 


75.    The  (sin~'x)"f7.T  can  l)e  integrated  by  the  aid  of  a  reduc- 
tion formula. 

Let  z  =  sin"^a;; 

then  a;  =  sin  2, 

dx  =  cosz.r/z, 

and  j  (siu^' a;) %/.<•=  iz"cosz.dz. 


68  INTEGRAL   CALCULUS.  [Art.  76. 

Let  ti  =  2", 

dv  =  coii z.dz ; 
da  —  )tz"~^  dZy 
V  =  sin  z  ; 
I  z"coHz.dz  =  z"s'u)z  —  )i  I  z'"'^ sin z.dz. 

i  z"~''s\uz.dz  can  be  reduced  in  the  same  way,  and  is  equal 
to  —  2"~'cos- -(-(n  —  1)  I  z^^'-iiosz.dz; 
hence 
i  z" cos z.dz  =  z"sinz -{■  11  z"~^cosz  —  n{)i  — I)  i  z"'^ cos z.dz,    [1] 

or  I  (sin"'.i-)"(/x-  =  .r(sin~'a-)"-f-  xVl  —  ^^^(sin"'^)""* 

-)>{)!  -\)  r(sin-'.r)"--fa-.  [2] 

If  >t  is  a  positive  integer,  this  will  enable  us  to  make  our  re- 
quired integral  dei)end  upon  |  dx  or  j  sin"'x.(/j:,  the  latter  of 
which  forms  has  been  found  in  (I.  Art.  81). 

Examples. 

(1)  Obtain  a  Ibrmula  for  I  ( vers "^r)" dr. 

(2)  r(sin-'.r)^/a;  =  .r[(sin-'.0'-  ^  •  3  .(sin-'.r)-+4  .3.2.1] 

-I-  4  Vl  -ar'sin-'.r[(sin-'a-)""'-  3  •  2]. 

7().  Integration  In'  substitution  is  .sometimes  a  valuable  method 
in  dealing  with  transcendental  forms,  and  in  the  case  of  the  trigo- 
nometric functions  often  enal)les  us  to  reduce  the  given  form  to 
an  algebraic  one.      Let  it  l)e  recpiired  to  tiud  |  (/  oina)  cosx.fZa;. 

Let  2  =  sin.c, 

dz  =  cosx.dx ; 


I  (  /sin.r)  eos.r  dx  —  i  fz.dz 


Chap.  VII.]  TRANSCENDENTAL    FUNCTIONS.  HO 

lu  the  same  way  we  see  tliat 
J  (/cosa;)sina;.(fa;  =-ifz.dz  if  z=cosj;, 

and 
J  [/(sin a-,  cos.r)]cos.T. r/.r=       j  [f(z,  Vl— 2-)]r/:r  if  z  =  s\n.r, 

j  [/(cos.r,  siu.r)]  sin.i'.(/.r  =  —  |  [f{z,   y/\—z-)']dz  if  2  =  cos  J, 
or.  111010  generally, 

rf{s\nx,  cosx)dx=       ( /(2.   Vl— 2-) — -^—         if  2  =  sin. c, 
-^  Vl-2- 

f  /(eosx,  sin.r)  (/.r  =  -  Cf{z,  VI-2-)  — ^^         if  2  =  cosx, 
J  J  ■  Vl  -z^ 


Since  any  trigonometric  fnnctiou  of  x  may  be  expivsseil  in 
terms  of  sin  a;  and  cos  a,-,  the  fornuilas  just  given  enable  us  to 
make  the  integration  of  any  trigonometric  function  depend  on 
the  integration  of  an  algebraic  function,  which,  however,  is 
frequently  complicated  by  the  presence  of  the  radical  Vl— z^. 

77.    A  better  substitution  than  that  of  the  last  article,  when 

the  form  to  be  treated  does  not  contain  s'xux  or  cos  a;  as  a  factor, 

.       X 

IS  2  =  tan-- 

2 

'II-       •  ^  2r?2 

1  his  gives  us  ax  = -, 

1  +2^ 

2z 

sin  X  = ■ , 

1  -+-  2- 

1  -2- 
cosa-= • : 

1+2^' 

whence     J/(sin x,  cos x)  dx  =  '2^^f(-=^,   T^y  7+?'    '" ^ ^ 

As  an  examjile,  let  us  find         |  '- 

J  a  +  b  cos  X 


70  INTKGUAL  CALCULUS.  [Akt.  78. 

Here  we  have 
r      dx       ^  2  r (h ^2  C  '^^ 

J  a  +  hcoBx        J  (1   t  ;,^)ra  I  h^-\\      J  a-\-h  +  {a-b)z' 

a  —  hJ  a  +  h      ^2       \lo?  —  b^  K^a  +  h     J 

a—h  by  I.  Art.  77,  Ex.  1. 

Hence    f '^ =     ,    ^        tap-^f^/^^- t.^n'^^'Y  if  a ->  ft. 

./a  +  ftcos.'c       Va^  — ft-  V\a  +  ft  2/ 

78.  I  sin"*xcos".r.f?a;  can  be  readily  found  by  the  method  of 
Art,  76  if  m  and  n  are  positive  integers,  and  if  either  of  them 
is  odd.     Let  n  be  odd,  then 

cos"  a;  =  cos"  ~^  a.*  cos  .r  =  (1  —  sin^a;)~5~cosa;, 
I  sin"'xc^s".T.f?.r  =  I  sin"" it" (1  —  siu^a;)~^cosa;.da;. 

Let  2  =  sin  a;, 

(h  =  cosiK.dr, 
I  sin" a;  coB'^x.dx  =  j  2;"'  (1  —  z^)~^dz^ 

which  can  be  expanded  into  an  algebraic  polynomial  and  inte- 
grated directly. 

If  ??i  and  n  are  positive  integers,  and  are  both  even, 

I  sin"a;  cos"a;.da;  =  l  sin".!-  (1  —  ^'nr xy^dx. 

sin"".!' (1  —  sin-.c)2  can  be  expanded  and  thus  integrated  by 
Art.  74  [1]. 

If  m  or  11  is  negative,  and  odd,  we  can  write 

cos"  .T  =  cos"  '  X  cos  a',     or     sin™  x  =  sin'""^^•  sin  x, 
and  reduce  the  function  to  be  integrated  to  a  rational  fraction 
by  the  substitution  of 

2  =  cos.r,     or     2  =  sin  a;. 
I  sin"*.!;  cos".c.f/.c  can  also  be  treated  by  the  aid  of  reduction 
fornuilas  easily  obtained. 


CiiAP.  VI I.]  TRANSCENDENTAL    FUNCTIONS.  71 

79.     I  tau"xdx  aud    I  — —  can  be  handled  by  the  methotls 

of  Art.  78,  but  they  can  be  simplified  greatly  by  a  reduction 
fonnula. 
We  have 

I  tau"a;.f/.i-=  j  tan"''-a;tan-.a'.rf.c=  j  tan"  - x  {sec' x  —  1)  clx 

=  I  tan"  -.rrf(tan.r)  —  |  tan"  -x.dXy 

/tan"  'r       /^ 
tan"x.dx  =  ~ '- —  I  tan"  ^x.dx;  [1] 

1         C  ^^'^    —  fsec^a;  —  tan-x  ,  ,  _  rd  (tanx)  _  r    dx 
J  tan".T     J         tan" a;  J     tan"x        J  tan^^V 

whence  (  — '- —  =  — — | —•  [2] 

J  tan"x  (?4  — l)tan"^x     J  tan"^aj 


Examples. 


sxv^xcos'  x.dx    — 


cos'".x*     cos°a; 


10  8 


,        2sin^x      2sin'x 

dx  = • 

3  7 


sin^x.fZ^  _  2cos'ig 


2cos^a;. 


(1)  /si 

(2)  I  cos^.x'  Vsinx*. 

.,s     r      2      -4      7  sin  ic  COS  x/sin^x      sin'aj      1\  ,    x 

(4)  j  cos'=«  s,o%.rfx-  =  ^^—(^-^  _____  -j  +  _. 

(.'))     I  =  sec. c  +  log  tan-- 

J  sinxcos^x  2 

(C)        -^^5 T-  =  seer  -  —r^  +  -  log  tan  -• 

(7)     f-^  = l_  +  _L_-f-logsinx. 

.'  tan*x  4tan*a;      2tan-x 


72 


INTEGRAL   CALCULUS. 


[Akt.  79. 


(8)f ^^-  =-4=log 

Ja  +  bcosx  sju^-a^        VM^  -  \ 


-Jh  +  a  +  ^h-a  .  tan^" 


h  —  a  .  tail  - 


4  +  0  tan  - 


(9)  (        ^^-^  =^tan->l 

J  0  +  4  sin  a;  3  \  i. 

(10)  r.       l"".     >    =Tlogsinf-logcos^^  +  jlog(3  +  2 
./ 3sina;  +  siu2a;      o  2  2      o 

...  V  /• cos  xdx 

^     V  (5  + 


o       sin.r 


4  cosx)^        9  5  +  4  cosx      27 


:tan  'frtan^ 


(12) 


fix 


+ 6  sin  x + c  cos  a;      Va-  —  6^ — c^ 


tan  ^ 


(a-c')tau';^  +  6 


(13)  Show  that  the  methods  described  in  Arts.  76-79  apply 
to  the  Hyperbolic  functions. 

(14)  I =    ,  tan- '  f  V  tanh-Uf^>a. 

V  a  +  ^  sinha-  +  <•( 


cosh  X 

2 
Vc«  -  «=>  -  A^ 


tan" 


(c  —  a)  tanh  o  +  ^ 
Vf-  -  «-  -  b'' 


fy^ 


1^ 


VIII.] 


DEFINITE    INTE(;itALS. 


73 


CHAPTER    VIII. 


D  E  F  I  N  I T  !•:     1  N  T  E  ( n I  A  L  S. 


80.  lu  I.  Art.  183,  a  dcfiuite  integral  has  boen  (Icliiiod  as  the 
limit  of  a  sum  of  infinitesimal  terms,  and  has  been  proved  e(iual 
to  the  difference  between  two  values  of  an  ordinary  inte<;ral. 

We  are  now  ready  to  put  our  definition  into  more  precise, 
and  at  the  same  time  more  general,  form. 

If  fx  is  finite,  continuous,  and  siiigle-valued  between  the 
values  x=a  and  x=  b,  and  we  form  the  sum 

(Xi  —  a)  fa  +  (x'a  —  a^i)  A  +(%—  X2)fXo-{ h  (.r„  ,-.(•„  .,)/.r„  j 

+  (^-.^•„  i)A,-i, 

where  rcj,  Xo,  ^s'-'^n-i  ^re  n  —  1  successive  values  of  x  lying 
between  a  and  b,  the  limit  approached  by  this  sum  as  n  is  in- 
definitely increased,  while  at  the  same  time  each  of  the  increments 
(Xi  —  a),  (Xo  —  Xj),  etc.,  is  made  to  ai)proach  zero,  is  the  definite 
integral  of  fx  from  a  to  b,  and  will  be  denoted  by  I   f.r.dv. 

If  we  construct  the  curve  y=fx  in  rectangular  co-ordinates, 
this  definition  clearly  requires  us  to  break  u[)  the  projt'ctiun  on 
the  axis  of  X  of  the  portion  of 
the  curve  between  the  points  .1 
and  B  into  n  intervals,  to  multi- 
ply each  interval  by  the  ordinate 
at  its  beginning,  and  to  take  the 
limit  of  the  sum  of  these  products 
as  each  interval  is  indefinitely  decreased  ;  that  is,  the  limit  of 
the  sum  of  the  small  rectangles  in  the  figure,  and  this  is  easily 
proved  to  be  the  area  ABAiBi- 

Now  the  area  ABAiB,.  found  l)y  the  method  of  I.  Chap.  V., 


i' 

A 

r- 

J 

U 

'U 

«i 

0 

a 

a 

, 

r-. 

r 

11 

)     ■ 

[/■^-"■■■].=. -[>•"■■■].■•• 


74  INTEGRAL   CALCULUS.  [Aur.  8L 

Therefore  r/:«-'^-^'  =  T  f/'-''"-^-'' 1       -     |!/a^.rf.r        .  [1] 

That   is,     I   fx.dx  is  the    increnieut   produced   in    j  fx.dx   by 

changing  x  from  a  to  b. 

It  is  to  be  noted  that  the  successive  increments  (a^i  — a), 
{x.i  —  Xi),  (0^3  — a'a),  etc.,  that  is,  the  successive  values  of  dx, 
are  not  necessarily  equal ;  and  also,  that  if  we  multiply  each 
interval,  not  by  the  ordinate  at  its  beginning,  but  by  an  ordinate 
erected  at  any  point  of  its  length,  the  limit  of  our  sum  will  be 
unaltered,     {v.  I.  Arts.  161,  149.) 

81.  It  is  instructive  to  find  a  few  definite  integrals  by  actu- 
ally performing  the  summation  suggested  in  the  definition 
(Art.  80),  and  then  finding  the  limit  of  the  sum. 

(a)  I    x.dx. 

Let  us  divide  the  interval  from  a  to  b  into  n  equal  parts,  and 
call  each  of  them  dx. 

Then  ndx  =b  —  a. 

Our  sum  is 
S  =  adx  +  (a+dx)  dx  +(«  +  2 dx)  dx  +  . . .  +  (a  +  (n -l)dx)  dx 

=  nadx  +  (1  +  2  +  3  H \-{n-l))dx^ 

since  ndx  =  b  —  a,  and  the  sum  of  the  arithmetical  progression 
H_.  +  ;3  +  ...4-(n-l)=^^i^. 

2                   ^  ^  2  2 

Hence  5  =  — : ^^ — : — -dx. 


Cirvr.  VIII.]  DEFINITK   INTEGRALS.  75 

As    we    increase   n   indefinitely,    dx    approaches    zero,    and 

J'*      ,   _    limit  ni-  —  a-      (b  —  a)  (Ix~\  _  b-      ir 
.  ^-^^-dr-oL     2  2        J"7~2' 

(b)  Ce^lx. 

Let  dx = 

n 

8  =  6"  dx  +  €"+'''  dx  +  6"+'^  dx  -\ h  6"+'"  '  "^  dx 

=  e" dx [  1  4- e*^  +  e'-'^'  +  e^   H f- e<"  '''^J  ; 

but   1  +  e"^  4- e-''^  +  •  •  •  +  e*"  ^^"^  is    a   geometrical   progression, 

and  its  sum  is 

g»d»  _  I   ^  gh-a  _   I 

e*""-!       „  7        /„.       ..X      f?-P 


Hence  *S  =  -^ •  e"  dx  =  (e"  -  e") 

and  J^  6="  dx  =  (e^  -  e")  ^^^^  ^  oL^^T^tJ  ' 

dx 
but  as  dx  approaches  zero,  -^ —  approaches  the  indeleiininato 

form  -  ;   but  since  the  true  value  of 
0 


r  e'da;  =  e''  — e". 
(c)  I    cos'^i 


s^x.dx. 

Let  dx  =  -,  and  let  n  be  an  odd  number. 
n 

Then 

S  =  dx  +  cos'  dx  •  dx  +  cos''  2  dx  •  dx  H h  cos''  ( n  -  2 )  dx  •  dx 

+  :os''  (n  —  l)  dx •  dx 
=  dx  +  cos^'dx  •  dx  +  cos"'  2  dx  •  dx  -| h  cos''  (tt  -  2  dx)  •  dx 

+  cos''  (tt  —  dx)  •  dx 
=  dx  +  cos"  dx  •  dx  +  cos''  2  dx  •  dx  H cos"  2  dx  •  dx 

—  cos"  dx-dx. 


76  INTKCtKAL   calculus.  [Art.  8L 

since  cos  (w  —  <f>)  =  —  cos  (}>. 

Hence  the  terms  cancel  in  pairs,  and  we  have  left 
S  =  dx 

and  JJcos'x.dx  =  J^^^  \^b^  =  0. 

(d)  I    sin- a;. da;. 

Let  d.r  =  — ,  and  let  n  be  an  odd  number. 
2)1 

S  =  sin'O-  (?.«'+sin^d.v-  (?.r+sin^2da;-(Z.r  -\ \- s'lu- {71  — 2)  dx-dx 

-f-sin^(?i  —  l)dx-dx 

=  shv'dx-dx  +  sin^  2dx-dx-\ f- sin-|  -  —  2dx  Vz.r + siii^[  -  —  dxpx 

=  sin^dic-c?x*-f  sin-  2dx-dx-\ |-cos-2f/.C'f/.r+cos-rfa;-fte, 

since  sin  |  ^  —  <^  j  =  cos  <^. 

Then  S  =  dx  +  dx  -j-dx---  =  ^^^  dx, 

since  sin^  ^  +  cos- <^  =  1 . 

Therefore        S  =- -, 

4       2 

and  I    sin-a;.dx  =  ^. 

(^>  i  T 

Here  it  is  best  to  divide  the  interval  between  o  and  b  into 
unequal  parts. 

Let  the  values  a;,,  x.,,  a*.,  •••  .t„_,  be  such  as  to  form  with  a  and 
b  a  geometrical  propfression. 


For  this  purjjose  take  7  =  \l   ,  so  lliat  a< 


b. 


Crap.  VIII.] 


DEFINITE    INTEGRALS. 


77 


Then  the  values  in  question  are  a^,  aq^,  aq^-'-aq"  ',  and  the 
intervals  are  a{q  —  1),  aq{q  —  1),  aq'^ {q  —  I)  •••  aq"  '  ( 7  —  1 ) , 
and  the  sum 

^^a(7-l)   ^  aq{q~l)   ^  (Uf  {g  -  I)   ^         ^  aq"  '  {q  -  I) 


a 
»(ry-l). 


aq 


aq" 


aq"  » 


To  prove  our  division  legitimate  we  have  only  to  show  tliat 
each  of  our  intervals,  a(g  — 1),  aq{q—l)  •••  o(y"-'(7.— 1 ), 
approaches  the  limit  zero  as  n  increases   indefinitely.      Since 

,.      h 
'^   =a 
the  limiting  value  of  q  as  )t  increases  must  be  1,  as  otherwise 
^'™'^  (/"  would  not  be  finite. 

Therefore       j^'^'^  [aq^q  -  1  )J  =  ^^f^  [aqHq  -  1 )]  =  0. 
We  have  then 


X 


''dx       limit  rcrn        limit  r    /         im       limitr    /         .x-. 


I  log- 
limit  a 

7  =  1  Ih^/ 


(7-1) 


n  log  q  =  log 


limit 
7  =  1 


log 


logr/^         ;]  ^*7  =  lLlogyJ  -a 

For       r^i  =rii=i. 

=  log/y  —  log  a. 
a; 


78  LNTEGKAL   CALCULUS.  [akt.  «2. 

Examples. 
(1)   Prove  by  the  methods  of  this  firticle  that 


jy 


log  a 

(2)  By  the  aid  of  the  trigonometric  formuhis 

co.s^  +  oos2^  +  cos3^H |-cos(;/  -  1)^ 

=  I    sin  nO  etn 1  —  eosji^  L 

siu^  +siu2^  +sin3^  H h  sin(ji-  1)0 

=  ^    (1  —  cos  71^)  etn sin?(6  L 

prove  that  |    cos.x'.d.x  =  s'mb  —sin a, 

and  I    sina'.cZa;  =  cos  a  — cos  6. 

(3)  Show  that     p'siu^r.c/x- =  0, 

and  that  |    cos-a;.(?.x'  = -• 

^0  2 

x'"dx  = — ,  usinsj  the  method  of 

»i  + 1 
Art.  81  (e). 

82.  "When  the  indefinite  integral  can  be  fonnd,  the  definite 
integial  |  fx.dx  can  usually  be  most  easily  obtained  by  era- 
ploying  the  formula  [1]  Art.  80,  and  this  can  always  be  done 
with  safety  when  fx  is  finite,  continuous,  and  single-valued 
between  x  =  a  and  x  =  b. 

Of  course,  if  the  indefinite  integral  is  a  multiple-valued  func- 
tion, we  must  choose  the  values  of  the  indefinite  integral  cor- 
responding to  X  =  (I  and  x  =  b,  so  tliat  they  may  be  ordiuates 

of  the  same  branch  of  the  curve  ?/ =   i  fx.dx. 


Chap.  VIII. 1  DEFINITE    INTEGRALS.  79 

Consider,    for   example,  |    -r-^^.-      The    indefinite    integral 
C   dr  »/-i  1  +a- 

I — ^,  =  tan  ^x    and    tan  'a;    is  a  multiple-vahied   function. 

Indeed,  y  =  tan  Kv  is  a  curve  consisting  of  an  infinite  number 

of  separate  branches  so  related  that  ordiuates  corresponding  to 

the  same  value  of  x  differ  by  multiples  of  tt.     On  the  branch 

which  passes  through  the  origin,  when  x=  —  \,  .v=?tan  'x=— ''; 

4 
on  the  same  branch,  when  x=l,  ?/ =  tan  \v=--     On  the  next 

o        i 
branch  above,  when  x  =  —l,  y=tan^x  =  —;  and  when  x=  1, 

y  =  — •     On  any  branch,  when  x  =  —  1 ,  y=  tan  ^x  =  —  -  +  mr; 

4  4 

and  on  the  same  branch,  when  x=  1,  y  =  -  +  »7r. 

Hence      f -^,  =  tan  '(1)  -  tan-H-l)  =  ^  +  ^  =  3.- 
Jil  +  .r  4      4      2 

X^     dx     _OTr      Stt 
il+a^"  4         4 

=  -  -\-  nir  —  I \-  7117     =  -• 

il+ar      4^  ^     4^       y      2 

By  I    fx.dx  we  mean  the  limit  approached  by    |  fx.dx  as  i 
is  indefinitely  increased. 

Examples. 
(1)  Work  the  examples  of  Art.  81  by  the  method  of  Art  82 

jr 

^is'mx.dx 


StT  TT 

~2' 


^H' 


=  V2-1. 


cos-x 


c/u    cr  +  ar      2  a 


''^  =iVa(V5-l). 


80  INTEGRAL   CALCULUS.  [Art.  83 

(o)  C  J"^-''    ^E  if  „  >  0.  .„„l  _  '^  if  a  <  0,  and  0  if  ,i  =  0. 
J»    a'  ■\-  X-      •>  2 

(6)  f  e  "rfx  =  ^    if  a>0. 

Jo  (I 

e'"  siii?Ha;.da;       =    , if  a>0. 


(8)  I     e~"^cosmx.dx       =  ^r-^ — r,  if  a  >  0. 


a-  +  ?u^ 


2  a;  cos  <^  +  ar      2  sin  <^ 


(10)  p ^ ^■. 

Jo    1  +  2  X  cos  <^  +  o;- 


sin< 


83.  When  fx  is  finite  and  single-valued  between  x  =  n  and 
x  =  b.  but  has  a  7t?ii7e  discontinuity  at  some  intermediate  value 
x  =  c 


j    fx.dx  =  I    /.c.(7.c  4-  I    fx.dx, 


and  tlierefore  i  fx.dx  can  be  found  by 
Art.  82  when  the  iudetinite  integral 
i  fx.dx  can  be  ()l)tained ;  but  when  fx  becomes  infinite  for 
x=a,  or  for  x  =  b,  or  for  some  intermediate  value  x  =  c, 
special  care  must  l>e  exercised,  and  some  special  investigation 
is  usually  required. 

fx.dx  approaches  a  finite 
limit  as  c  approaches  zero,  this  limit  is  what  we  shall  mean  by 

X  fx.dx;  if  I  fx.dx  increases  indefinitely  as  e  approaches 
zero,  we  shall  say  that  I  fx.dx  is  infinite;  and  if  |  fx.dx 
neither  :iii[)io:i(hes  a  finite  limit  nor  incieases  indefinitelv  as  « 


Chap.  VIII.]  DEFINITE   INTEGRALS.  81 

appronclu's  zero,  wc  shall  say  that    |   fx.dx  is  indeterminate. 

fx.dx  can  be  safely  employed 
in  mathematical  work. 

If  J'x  is  iutiuite  when  x  =  b  and  j  fx.dx  approaches  a  finite 
limit  as  e  approaches  zero,  that  limit  is  the  valne  of  I   fx.dx. 

If  fx  is  infinite  when  x  =  c,  and  each  of  the  expressions 
1  fx.dx  and  I  /.l^f?.^•  approaches  a  finite  limit  as  c  approaches 
zero,  the  snm  of  these  limits  is  I  fx.dx.  Should  eitiier  or 
both  of  the  expressions, 

X fx.dx,         I   fx.dx, 

fail  to  approach  a  finite  limit  as  e  approaches  zero,  I  fx.dx  is 
either  infinite  or  indeterminate,  and  cannot  be  safely  used. 

When  the  indefinite  integral  of  fx.dx  can  be  obtained  there 
is  little  ditliculty  in  deciding  on  the  nature  of  I  fx.dx  in  any 
of  the  cases  just  considered,  or  in  getting  its  value  when  that 
value  is  finite  and  determinate. 

For  example, 

Ji'^dv 
\    —  is  infinite,  since 
U      X 

I  —  =  log  X  and    |     —  =  log  ( 1 )  —  log  e  =  log  -, 

J     X  °  J,      X  ^^     '  € 

and  increases  indefinitely  as  e  appinaches  zero. 
dx 


<^>  i  r 
/ 


not  finite  and  determinate,  for 


.T* 


dx       _^j^^^^.l+a; 


1  —  a;-      "     '  \  —X 

Jo     ]  -a-       -     ■   \     e     J 
and  increases  indefinitely  as  e  approiu-lies  zero. 


82  INTEGRAL  CALCULUS.  [Art.  84. 

(c)      I  -  is  finite  and  determiuate,  for 


/ 

Jo    ^Hfi 


sld'- 

x^ 

dx 

Va^- 

ar 

'  '   dx 

.       iX 


sin"^  '-^^ — -  —  sin~^  0 


y/aF-x"  «  " 

and  its  limiting  value  as  c  approaches  zero  is  sin-'(l)   or  |- 

— ^ r  is  finite  and  determinate,  for 

0    (l-x-)» 

Jo  (Tir^i-^'    ^'    ^^^' 

and  its  limiting  value  as  e  approaches  zero  is  f  —  |. 

and  its  limiting  value  as  c  approaches  zero  is  —  |  —  f ,  and 
consequently 


Ji    (l-x)^      -       ■'       "^       '^ 


84.  When,  as  is  sometimes  the  case,  the  indefinite  integral 
cannot  be  obtained  and  the  function  to  be  integrated  becomes 
infinite  at  or  between  the  limits  of  integration,  we  have 
recourse  to  a  very  simple  test  which  is  easily  obtained  by 
the  aid  of  the  following  important  theorem,  known  as  the 
Maxinmm-Minimum  Theorem. 

If  a  given  function  of  x  is  the  product  of  ttvo  functions  both 
finite,  continuous,  and  single-valued,  one  of  which  <f>(x)  does  not 
change  its  sign  between  x  :=  a  and  x  ^  b,  and  if  M  is  algebra- 


Chap.  Vlll.]  DEFINITE    INTEGKALS.  83 

ically  the  greatest  and  m  the  least  value  of  the  other  factor 
f(x)   between    x  =  a   and   x  =  b,  J    f  (x)  <^  (x)  dx  lies  between 

M   I     <^(x)dx  and  m    (   <^(x)dx. 

To  prove  this  theorem,  let  us  first  suppose  tliat  «^(.r)  is 
positive  between  x=^a  and  x  =  b.  M  —  f(x)  is  positive  for 
the  values  of  x  in  question,  [M  — /(^)]  <l>  (a-)  is  positive,  and, 
therefore, 

f\M-f(x)]<f>(.r)dx>0 

and  ^^Sy  (•'■)  '^■''  >  X'-'"'^'''^  '^  '^■'■^  '^■^*  ^^^ 

/(a-)  —  ??i  is  positive  for  all  values  of  x  between  x  =  a  and 
x  =  b,  [f(x)  —  ?n]  <f)  (x)  is  positive,  and,  therefore, 

f\f(^-)-m^4>(^)dx>0 
and  C  f(x)<f>  (x)  dx  >  m    C  cj>  (x)  dx.  (2) 

f(x)  (f>  (x)  dx  lies  between  M  I   4>  (x)  dx  and  7n 

I   <^  (x)  dx.     It  is  easy  to  modify  this  proof  to  meet  the  case 

where  <^  (x)  is  negative. 

We  can  briefly  formulate  the  result  of  the  Maximum-Mini- 
mum Theorem  as  follows  : 

fy(x)  <t>  (x)  dx  =f($)  jy  (x)  dx,       (3) 

where  i  is  some  value  of  x  between  a  and  b. 

Let  us  apply  this  theorem  to  the  consideration  of  I  f(x)dx 
when/(a;)  becomes  infinite  for  x^a. 


84  INTEGRAL   CALCULUS.  [Art.  84. 

lu  order  tliat  ,'       1  ./'(•'')  ff^      should  be  finite  and  de- 

terminate  it  is  easily  seen  to  be  necessary  and  sufficient  that 

limit  r  limit  /  C'^lw  n  /  M     u     i -.  . 

II      JK-'')'!^']      should  be  equal  to  zero. 

Let  us  write  /(.r)  in  the  form  (■^•~  ^0\/'(a-)  ^^^^  ^^.^  0  <  /:  <  1. 
^  ^  (x  —  ay 

-7  is  positive  for  all  values  of  x  greater  than  a. 

(x  —  up 

=tf-«)'/(f)£:-^. 


cl-* 1-t 


-  limit 
,  limit 


£j(^)  d-r'j  =  a  -  ay-fd)  ^. ;   «  <  ^  <  a  +  c  ; 

does  not  increase  indefinitely  as  $  approaches  a. 

Calll^  —  dtri,  whence  $  =  a  -\-  rj.  Then  a  sufficient  condition 
that  I  fix)  dx  shall  be  finite  and  determinate  when  f(a)  =  oc 
is  that  Tff{(i  +  rf)  shall  not  increase  indefinitely  as  rj  ap- 
proaches zero,  0  <  A-  <  1.     If  we  write  f(x)  =  ^^ )J  \  ) 

and  proceed  as  above,  we  can  sliow  that  a  necessary  condition 
that  I  f{x)  dx  shall  be  finite  and  determinate  when  f(a)  =  oo 
.     limit  r    ^/     I     NT      A 


ClIAP.    VIII.] 


DEFINITK    INTKCIIALS. 


85 


If /(/>)  =  GO  our  sufficient  condition  is  tliat  rf'/ij)  —  -q)  shall 
not  increase  indefinitely  as  17  d=  0,  0  <  A-  <  1 ;  and  it. /'('')  =  <» 
that  neither  ri\f{c  —  rj)  nor  rff{c  -\-  rj)  shall  increase  indefi- 
nitely as  77  =  0,  0  <  ^-  <  1. 

Let  us  apply  our  tests  to  the  examples  considered  in  Art.  83. 

rUlx  .  limit  Ft?"!       , 

(d)        I     — =:x  because  ,     -     =1. 

X*     ff.r 
-^  is  indeterminate,  for 

t  r       V      ~\  _  ^i"iit  r_i "I  _ 


limit 


and 


lim 
V 


mit  r  T]  ~\ limit  F  —  1  ~|  _  _  . 

=  Oil  -  {rj+  If  j-  y  .^Ol'l^y]-        *• 

J""       dx 
—    '  is  finite  and  determinate,  for 


\'^ 


:Oif^</.<l. 


^      X(J.r 


is  finite  and  determinate,  for 


id 


86  INTEGRAL  CALCULUS.  [Art.  84. 

Even  when,  as  in  the  examples  just  given,  the  indefinite 
integral  can  be  obtained,  there  is  a  decided  advantage  iu  using 
the  very  simple  method  of  this  article.  For  if  the  application 
of  the  test  shows  that  the  definite  integral  in  question  is  infinite 
or  indeterminate,  the  labor  of  finding  the  indefinite  integral  is 
saved  ;  and  if  the  application  of  the  test  proves  the  definite 
integral  finite  and  determinate,  it  follows  that  the  indefinite 
integral  does  not  become  infinite  for  the  value  of  x  which 
makes  the  given  function  infinite,  and  consequently  when  the 
indefinite  integral  has  been  obtained,  the  method  of  Art.  82 
can  be  used  without  hesitation. 

As  an  example,  wliere  the  indefinite  integral  cannot  be  ob- 
tained, let  us  consider  at  some  length 


s: 


los:    )  dx. 


If  71  is  positive,  [  log    )    is  continuous  and  single-valued  be- 


tween x=0  and  x=  I,  but  becomes  infinite  when  x=0.  We 
must  then  investigate  the  limiting  value  of  t;*'  f  log  -  j  as  77 
approaches  zero. 


y  found 
fractional.      For   positive   values  of  ?(,    |    [log    )  dx  is,  then. 


r)"  I  log  -  I  is  indeterminate  when  r)=0,  but  its  true  value  is 
easily  found  to  be  zero  if  11  is  positive,  whether  ?i  is  whole  or 


finite  and  determinate 

If  n  is  negative,  call  n  =  —  m. 

Then 


is  continuous   and  single-valued  from  x=0  to  x  =  \,  but  be- 
comes infinite  when  «=  1. 


Chap.  VIII.] 

We  must,  then,  find 


DEFINITE    INTEGRALS. 

limit 


87 


hich  proves  to  be  0  if  m  <!  1 ;  if  m  =  1, 


L('-i^,)' 


=  1; 


and  if  ni  >  1,  it  is  infinite. 


is,  then,  finite  and  determinate  if  m  <,  1,  but  infinite  if  m  =  1 
or  ?>i  >  1 ;  and  we  reach  the  result  that 


r('»4)' 


dx 


is  finite  and  determinate  if  7i>—  1,  but  infinite  if  ?i  =  —  1  or 
?i<— 1. 

Examples. 

(1)  Prove  that 

C}^.cU.  fi^.do.,  r^iogfi±^\ 

Jo   \-x        '  Jo   l-or'         '  Jo    a;     \\-x)' 
are  finite  and  determinate. 

(2)  Prove  that 

J^^    ,     C.^1^1 ,  where  m  and  ?i  are  inte<rer8,  and 
0    \  —X*     Jo    1  —  X-" 

J^»  ^  1 
.  rfx',  are  not  finite  and  (letormiiiate. 
u     \  —X 

/•I 

(3)  Find   for  what  values  of  n    \     (log.c)"'/.':    is  tiiiitc  miuI 
determinate. 


88  INTEGRAL   CALCULUS.  [Art.  85 

(4)  Find  for  what  values  of  m  and  n    |    a,-"'nog- j  dx    is 
finite  and  determinate. 

(5)  Show  that    |    af~\'l  —x)''^^dx  is  finite  and  determinate 
if  m  and  n  are  positive. 

(6)  Prove  that    |     logsinx'.dr  is  finite  and  determinate. 

(7)  Show  that  the  following  integrals  are  finite  and  deter- 
minate, and  obtain  their  values  : 


r-    dx     ^' 

Jo    Vu-  -  .r 


s: 


dx 


r- 


\l  ox  —  if- 
dx 


V.ir  -  1      3 


85.    It  was  stated  in  Art.  >*■•!  that  l)v    |  fx.dx  we  mean  the 

fx.dx  as  b  is  indefinitely  increased,  and, 
/^ 
as  we  have  seen,  if  the  indefinite  integral  I  fx.dx  can  be  found, 

there  is  no  difficulty  in  investigating  the  nature  of    |  fx.dx  and 

in  obtaining  its  value  if  it  is  finite  and  determinate.     There  are, 

however,  many  exceedingly  important  definite  integrals  of  the 

form    I  fx.dx  whose  values  are  obtained  by  ingenious  devices 

without  employing   the   indefinite   integral,    and   these   devices 

are  valid  only  piovided  that  the  integral  in  question  is  finite  and 

(leleruiiniiti',  since  an  infinite  value  not  recoy;nized  and  treated 


Cn.vr.  Viri.j 


DEFINITE    INTEGRALS. 


89 


as  such,  or  a  value  absolutely  indeterminate,  renders  inconclu- 
sive any  piece  of  mathematical  reasoning  into  which  it  enters. 
If  we  construct  the  curve 

y  =/r,  I  Jx.dx  is  the  limit- 
ing value  approached  hy  the 
area  ABB^A^,  as  OB^  is  in- 
definitely increased  ;  and  in 
order  that  this  area  should 

be  finite  and  determinate,  it  is  clearly  necessary  and  sufficic]it 
that  the  area  BCCiBi  should  approach  zero  as  its  limit  as 
first  OCi  and  then  OB^  is  indefinitely  increased. 


That  is, 


limit 


:tC=t[x>'-*])=«- 


86.    A  sufficient  condition  that 
limit  r  limit 
i  ^  oc 


\JrL{S>y-)\='' 


can  be  easily  obtained  by  the  aid  of  the  Maxivmm-Mlnimum 
Theorem  (Art.  84). 

Let /(a-)  be  single-valued  and  continuous. 


We  can  write /(x)  in  the  form 


k>\\  then  by  (3) 


Art.  84. 


iiid 


limit  r  f  V   w  "1       ->'<'^^)     1      J  ^t 


1 


not  increase  indefinitely  as  ^  increases  indefinitely. 


90  INTEGRAL   CALCULUS.  [Art.  86. 

If,  then,  [icy(a*)]^=^  is  not  infinite,  k>l, 

I    f(x)dx  is  finite  and  determinate, 
(a)   As  an  example  of  the  use  of  this  test  we  will  prove 


X" 


e~^^dx  finite  and  determinate. 


e   ^'  is  single- valued,  finite,  and  continuous  for  all  values  of  x. 
limit 

X 


x^'e  ^°    ,  A;  >  1,  is  easily  found  and  proves  to  be  zero. 

Hence,  I     e-'^'dx  is  finite  and  determinate. 

(b)   Let  us  consider  f      dx. 

Jo  X 

sin  ax  .  .  . 

IS  equal  to  a  when  x  =  0,  and  is  finite,  continuous, 

and  single-valued  for  all  values  of  x. 
Let  a  be  a  given  constant ;  then 

Jf'^sinaa-,          fs'max^     ,     r^^  sin  ax, 
dx=  I     f/x-j-  I      dx, 
ox                    Jis           X                    Jo.           X 

and  I     dx  is  finite  and  determinate. 

Jo         x 

By  integration  by  paHs. 

/sin  ax  .             cos  ax      If  cos  ax  , 
dx  = I   5—  dx, 
X                       ax         a  J       x^ 

X"  sin  ax    ,         cos  aa       1    T*  cos  ax    , 
dx  = I      — -—  •  dx, 
X                    aa          a  J  a.        x^ 

,  C^  cos  ax  -    .     _    .  ,   - 

ana  I     — -r-dx  is  finite  and  determinate  since 

Ja         x^ 
limit  Fa;* cos  ax~\ _   limit  rcosn'.r~|_      c  i  ^  ,  ^  r, 
a;  =  oo  |_       x^      J      a-  =  oo  L  ^        J 

(r)     i     cos  (x^)  dx  is  finite  and  determinate,  for  cos  (x^  is 
finite,  continuous,  and  single-valued  for  all  values  of  x,  hence, 


Chap.  VIII.]  DEFINITE    INTEGHALS.  91 

j     COS  {x^)dx  is  finite  and  determinate  ;  and 

-         r^  sin(.r-)f/.r  .     „    . 
and      J^    ^^^^ —  IS  finite  and  determinate  since 

limit  Fx^' sin  (.r-) n 

^    =0.     I  <A-<2. 

a:  =  00  L        x^       J 

Examples. 

(1)  Construct  the  curves  ?/  =  e  '-;  y  =  '- — -^  ;  ?/  =  cos(dr). 

(2)  Prove  that  the  following  integrals  are  finite  and  deter- 
minate : 

fc/o       ar  ./o     -vy^  Jo  X 

e~"-'"coste,rf.r,        I     e"<"'x^".fZ.c,  )    e  "  ^-.dx^ 

0  »/0  ^0 

(3)  Show  that  |  x^e'^.dx  is  finite  and  determinate  for  all 
values  of  n  greater  than  —  1 . 

87.  When  we  have  occasion  to  use  a  reduction  formula  in 
finding  the  value  of  a  definite  integral,  it  is  often  worth  while 
to  substitute  the  limits  of  integration  in  the  general  foriinila 
before  attempting  to  apply  it  to  the  particular  problem. 


For  example,  let  us  find    |       '   ' 

Jo  V«-  —  x^ 

We  can  reduce  the  exponent  of  x  by  [t].  Art.  01, 

Cx-  ^z''dr  =  -^  "'"^^    _a(m-u)    r.  ..  ,., 
J  b  (m  +  njy)      b  {m  +  »;>)  J 


92  INTEGRAL   CALCULUS.  [Art.  87. 

For  our  example  this  becomes 

J  ^  '  -m+1  -m  +  lJ 

When  X  =  0,  aud  also  wheu  x  =  o,  ' ^ '-  =  0, 

Hence 

f>-i(a2_  x-^r'dx  =  ^^^""'  ~  ^^  fa""  X"'  -  •^•')~^cZx ; 
»/o  m  —  1    »/o 

J" V(a2  _  a^)  -i  da:    =  -  .  a^  f  V (a-  -  ar )"^ da; 
0  6         »/o 

=  5.^.^4  rV(a2_.r^)-4daJ 
6     4        Jo 

=  5.§.1.„.  f    rf-'' 

6     4     2         Jo 

r 


Therefore 


Vu-  —  X"^ 
ic^dx         1     3     5     Tta^ 


Va--x-      -2     4     6 


Examples. 


x'dx  -  .  -  a* 


Jo   V^^S^r^  3     0 

2)  I    \!a-  —  x-'dx      =^- 
c/o  4 

3)  C^  yf^f^^ .  df  =  1  .  ''^-' 
Jti  4       4 

4)  j;v(„=_a-y.*.=i.i 


1    -^ 

7ra". 


^.     C'i   .   „      ,  \:?>.h  ..An  —  1)     TT     , 

5)     I    siu  x.ax  =— ^-  •      when  ?i  IS  even 

'  Jo  2.4.6...?t  2 

= ^^ wlien  n  is  odd. 

3.5.7... n 


Chap.  VIII.]  DEFINITE   INTEGRALS.  93 

n  IT 

(6)   Show  that  |     cos"a'.(Zx=  |  '^\n"x.dx. 

r\  a?"dx   ^1.3. 5. ..(271-1)     TT 
^''  Jo  Vn^'  2. 4.6. ..2)1        "  2* 

Suggestion :  let  x  =  sin  ^ 

^   ^  Jo   VI^T^'      3.5.7. ..(2u  +  l)" 

(9)   From  Exs.  7  and  8  obtain  Wallis's  formula 

n      2.2.4.4.6.6.8.8... 
2~~  1.3.3.5.5.7.7.9...' 

r^  x^"dx        r*  xr"-^^dx 

Vl-ar^ 


Suqqestion :    I    — =zz=  >  I       ,  >  I    —;=■ 

Jo  Vl-ar^     »^o  Vl-ar     »^«  Vl 


88.    When  in  finding  i    fx.dx  the  method  of  integration  by 

substitution  is  used,  and  y=Fx  is  introduced  in  place  of  x,  we 
can  regard  the  new  integral  as  a  definite  integral,  the  limits  of 
integration  being  Fa  and  Fb,  and  thus  avoid  the  labor  of  re- 
placing y  by  its  value  in  terms  of  x  in  the  result  of  the  indefinite 
integration. 

Let  us  find  f  e'"Vl  -  e"''"  •  dx. 

Substitute  y=e". 

dy  =  ae'^dx. 

Hence  C  e"  Vl  —  e-" .  dx  =  ^  j  V 1  -  y- .  dy. 

When  x  =  — 00,  ^  =  0,  and  when  x  =  0,  y=\. 

Therefore     fV'Vl-e^^" . rfx  =  1  f  Vl-/ •  dy  =  -^. 
J-oo  <^.'o  4  a 


94  INTEGRAL   CALCULUS.  [Art.  88. 

There  is  one  class  of  cases  where  special  care  is  needed  in 
using  the  method  just  described.  It  is  when  y  has  a  maximum 
or  a  minimum  value  between  x  =  a  and  x=^b,  say  for  x=:c, 
and  X  is  consequently  a  multiple-valued  function  of  ?/. 

For  suppose  1/  a  maximum  when  x  =  c,  then  as  x  increases 
from  a  to  b,  y  increases  to  the  value  Fc,  and  then  decreases 
to  the  value  Fb,  instead  of  simply  increasing  or  decreasing 
from  Fa  to  Fb.  If  <fi(y)dy  is  the  result  of  substituting 
y  for  X  in  fx.dx,  <j>y  is  a  multiple- valued  function  of  y, 
and  it  will  always  happen  that  when  y  passes  through  a 
maximum,  we  pass  from  one  set  of  values  of  x  to  another, 
and  therefore  from  one  set  of  values  of  <f>y  to  another,  and 
in  that  case  it  is  necessary  to  express  our  required  integral 

<j>y.dy  +  I  <t>y.dy,  taking  pains  to  select  the  correct  set  of 

Fa  ^Fc 

values  for  <f>y  in  each  integral. 

If  y  is  a  minimum  between  x  =  a  and  x  =  b,  essentially  the 
same  reasoning  holds  good. 

A  couple  of  examples  will  make  this  clearer. 


(a)     Take  C 


x.dx 


V2  ax  —  x' 

Let  ?/  =  2  ax  -  x^.     Then  ^  =  2  (a  -  x)  =  0  when  x  =  a. 
dx 

— ^  =  —  2,  and  y  is  a  maximum  when  x  =  a. 
dx"  ^  

X  =  a  ±  Va-  —  ?/, 

dx=:^-^y^- 


2V«^ 


Since  -'-  is  positive  from  x  =  0  to  x  =  a,  and  negative  from 
dx 

x=a   to   a;=2a,    dx= -JL=    fi»d   x  =  n  —  yjd-  —  y    from 

2  Va^  —  y 


dy 


x  =  0   to    x  =  a,    and    dx  = ■         ,  and  x  =  a  -\-  V(r  —  y 

2  Vtt^  —  y 
from  x  =  a  to  x  =  2  a. 


CiiAi-.  VIII.]  DEFINITE    lNTE(;itALS.  95 

Hence 


Jp"'    xdx       _  r"     xdx  r-"    xdx 

"  V2  ax  —  ar     ^^  V2  ax  —  ar'     •^''  V2  ax  — 


Very  —  y-  '^^''-     \ld-y  —  y^ 

2^/0  yja^y^y^  2»/U         yja^y^y^ 

=  r°'   ^^^       =7rft.  (Ex.  7,  Alt.  84) 

*^°\Jd-y  —  y- 

«/o(sina;  +  cosa;)^ 

Let  V  =  sin .);  -|-  cos  x.     -^  =  cos  a;  —  sin  .r  =  0  wliou  x  =  -• 
da;  4 

^=  -  siu.r  —  cosa;  =  — V2  when  x  =  --      Tlierefore  y  has 
dx'  _  ^4 

a  maxiimim  value  \/2  wlien  x  =  ^. 

4 

2/  =  sin.r +  cosx=  V2  .  cosf  j  —  x\ 

y  ,7.._^      f^.V 


cos^-^,  dx  =  ± 


4  V2  V2  -  f 

Since  ^  =  0  and  ^^"-'^  <  0  when  x-=-,  it  follows  that  -^  is 
positive  from  x  =  0  to  -J-'  =  p  and  negative  from  .v  =  -  to  x  =  • 
Hence  we  have 

»  IT  » 

X^  da;  ^  ri__^^i__  4.  r!__J?!L_ 

(sin.i;  +  cosa;)2     Jo  (siux  +  cosx)-     J„(siua; +  cosj;)' 

4 

J-^v^       dy        _   /-'         dy         _  g  f^^        '^■' 


96  INTEGRAL    CALCULUS.  [Art.  80. 

Let  -^  =  sin^; 

V2 

IT 

0  (siux  +  cosic/ 


and 


u  (sm.i-  + 


EXAMPLE. 

dx 


=  00. 

cos  .c)- 


89.    Differentiation  of  a  definite  integral. 

We  have  seen  in  Art.  51  that  a  definite  integral  is  a  function 

of  the  limits  of  integration^  and  not  of  the  variable  with  respect 

to  which  we  integrate  ;  that  is,  that   |  fx.dx  is  a  function  of  a 

and  Z>,  and  not  a  function  of  x.     Strictly  speaking,    I  fxAx  is 

a  function  of  a  and  &,  and  of  any  constants  that  fx  may  con^ 
tain,  where  by  constant  we  mean  any  quantity  that  is  indepen- 
dent of  X. 

If  the  limits  a  and  b  are  variables,  they  are  always  indepen- 
dent of  the  X  with  respect  to  which  the  integration  is  performed, 
which  must  from  the  nature  of  the  case  disappear  when  the 
definite  integral  is  formed,  as  it  always  ma}'  be  in  theory,  from 
the  indefinite  integral ;  and  this  assertion  holds  good  even  when 
the  same  letter  which  is  used  for  the  variable  with  respect  to 
which  the  integration  is  performed  appears  explicitly  in  the 
limits  of  integration. 

Thus  if  we  write    I  sin.r.r?.r,  the  x  in  sina;.c?.r  and  the  x  which 

is  the  upper  limit  of  integration  do  not  represent  the  same 
variable,  and  are  entirely  unconnected.     Indeed,  the  former  x 


Cn.vi'.  VIII. ] 


DEFINITE    INTEGRALS. 


07 


may  be  replaced  by  any  other  letter  without  affecting  tiie 
value  of  the  integral.     For 

Xs'inx.dx 

=  1  —  cos  X. 

Let  us  now  consider  the  possibility  of  differentiating  a 
definite  integral. 

Required  i)„  |   f(x,  a)dx,  where  a  is    independent  of  x, 

and  a  and  b  do  not  depend  upon  a,  and  D^f{x,  a)  is  a  finite 
continuous  function  of  a  for  all  values  of  x  between  a 
and  h. 


We  have 
n£f(x,a)dx  = 


limit 


Cf(.l\  a  -\-  \a)d.r  -ffi.-'-^  ")'^-' 


Aa 


limit 
Aa  =  0 


^  PY  lilllit    rf(.r,  a  +  Aa)  —/(■»•.  a)"[  \  ^^  . 

Hence,         ^-f/i^'  0 '^-^  =X' '^^"'^*^"^'  ''^-' ''■^'  ^^  -* 

and  we  find  that  we  have  merely  to  differentiate  uiuUt  the 
sign  of  integration. 


98  INTEGRAL  CALCULUS.  [Art.  89. 

If  Daf(x,  a)  becomes  infinite  for  some  value  of  x  between 
a  and  b,  or  if  one  of  the  limits  of  integration  is  infinite,  the 
proof  just  given  ceases  to  be  conclusive  and  [1]  must  not  be 
assumed  to  hold  good. 

The  truth  of  the  converse  of  the  proposition  formulated 
in  [1]  can  be  easily  established  by  differentiation,  and  we 
have 

J    M  /(a;,  a)t?a;  Jc?a 

or  even 

£'\^£nx,a)dx~jda 

=£[_£' f(x,  a)  da^dx,  [3] 

if  a,  b,  c,  and  d  are  entirely  independent. 

[2]  and  [3]  are  of  course  subject  to  limitations  easily  in- 
ferred from  the  limitations  on  [1],  stated  above. 

If,  however,  in  [3]  b  is  infinite,  it  can  be  shown  by  the  aid 
of  the  Maximum-Minimum  Theorem  that  a  sufficient  condition 
that  [3]  should  liold  good  is  that  it  shall  be  possible  to  find  a 
value  of  X  such  that  for  that  value  and  for  all  greater  values 
x^'f(x,  a)  shall  be  less  than  some  fixed  value  for  all  values  of 
a  between  c  and  d,  k  being  greater  than  1. 

If  d  and  b  are  both  infinite  there  is  also  the  corresponding 
condition  involving  x^f{x,  a). 

We  are  now  able  to  state  a  sufficient  condition  that  [1] 
shall  hold  when  b  is  infinite.  It  is  that  it  shall  be  possible 
to  find  a  value  of  x  such  that  for  that  value  and  for  all  greater 
values  x^'Daf{x,  a)  shall  be  less  than  some  fixed  value. 

Suppose  now  that  we  are  dealing  with  variable  limits  of 
integration. 


CuAr.  Vlli.J  DEFINITE    INTKCIiALS.  99 

Let  lis  find  first  —  I  fx.dx. 

rJzJ.. 


Let  C fx.dx  =  Fx,  then  i  \fx.dx  =  Fz  -  Fa  ;    r 


and   since    bv 


definition  ^"  =  A',  it  follows  that  —=fz. 

dx      "  dz 


Hence  ±  Cp.,,.  =  'll^^^I^=fi.  [4] 

dzJa  dz  •  "-  -^ 

In  the  same  way  it  may  l)e  shown  that 

|J>..<.-  =  -,i.  [5] 

Let  us  now  take  the  most  complicated  case,  namely,  to  find 

d   C^ 
—  I  /(a;,  a)  dx,  where  a  and  h  are  functions  of  u. 

da  J  a. 

Let  f/C^,  ")  d.i;  =  F(^x,  a) ; 

then  u  =  r/(.r,  a)  dx  =  F{b,  a)  -  F{a,  a) , 

da  da  (Za 

but  as  b  and  a  are  functions  of  a, 

,  dF(a,a)       »-,  77,,       ^  dff  ,    TV   7-./       N 

and  — ^--^— ^  =  Z)„i^(a,  a)  - +  />„i=^(a,a), 

da  du 

by  L  Art.  200. 

D,F(b,a)=f{b,a), 

D„  F{a,  a)  =f{a,  a). 
Hence  '^"  =  D.  lF{b,  a)  -  F(a,  a)]  +/(6,  a)  'f'  -./( .«,  a)'^, 

da  d,i  (/a 


-f  C"f(x,  a)  dx  =    r(Z>„/(.r,  a))  dx  -\-f{b,  a)  f  -/(.(,  a)  '^^^    [H] 
aa«/a  «/«  Ott  Utt 


100  INTEGRAL    CALCULUS.  [AliT.  91 


Examples. 

■» 

siu  (x  +  ?/)  dx  =  (a;  -f- 1 )  sin  {xy  +  y)  —  ainy. 


(1)  Af 

(2)  —  Cx^(1x  =  - 
^    .    clxjo  3 

(3)  ^JVT^ 

90.  When  the  indelinite  integral  cannot  be  found,  the  prob- 
lem of  obtaining  the  value  of  the  definite  integral  usually  be- 
comes a  more  or  less  difficult  mathematical  puzzle,  which  can 
be  solved,  if  solved  at  all,  only  by  the  exercise  of  great  inge- 
nuity. iSome  of  the  results  arrived  at,  however,  are  so  impor- 
tant, and  some  of  the  devices  employed  so  interesting,  that  we 
shall  present  them  briefly  here.  But  we  must  repeat  the  warning 
that  most  of  the  methods  are  valid  only  in  case  the  definite 
integral  is  finite  and  determinate  ;  and  erroneous  results  have 
more  than  once  been  obtained  and  published  when  a  little  atten- 
tion to  the  precautions  described  in  Articles  83-86  would  have 
prevented  the  mistake. 

91.  Integration  by  development  in  series. 

(a)     f'i^^.dx.  (v.  Art.  84,  Ex.  1.) 

Jo     I  —X 

,J_         =  (1  -  .r)-'  =  1  +  .'•  +  .r  -f  -r'  +  •..,  if  .r  <  1 . 
1  — a; 

— M —  (Jx  =  j   ( log  X  -\-  X  log  X  +  .r  lou  a'  +  •  •  • )  dx. 

0     I  —  X  Ju 

Jx"\orrx.dx  = -•  (v.  Art.  55  (a).) 

\,         ^  (»  +  l)' 

Therefore 

Jn    1  -X  >^l-^2-'      ;5-      4'  /  6 

(v.  Todhuntcr's  Trigonometry,  Chap.  XXIII.,  Ex.  1.) 


Chap.  VIII.]  DEFINITE   INTEGRALS.  101 

(6)  r  log/'^^liiV/x".  (v.  Art.  Ht;,  Kx.  L'.) 

^''^  (J^t)  =  ^*'-  (r^')  =  i^^g  ( 1 + « ')  -  i"s^'  ( 1  -  ^  ') 

~^  '         2  3  4   ■"       V     '^  2  3  4   ■■/ 

(I.  Art.  130.) 
Hence 


r-(?^)-=^'re-^^T-9'- 


dx 


=  2(1+1  +  1  +  1  + 
o-      o-       < - 


But  i,  +  l+i  +  i  +  ...=^ 

1-       A-       :)-       f  8 


Therefore 


(v.  Tod hun tor's  Trio-.,  (liap.  XXIIl.,  Ex.  1.) 

+  1 


i  -<^  -=r 


(1)  1 1^ 


^^^s-^'  rfx 


Examples. 


12 


(4) 


1-3Vm 


if  A-<  1. 


102  INTEGRAL   CALCULUS.                        [Art.  92 

92.  Integration  by  ingenious  devices. 

(a)  C'logs'mx.dx.                                    (v.  Art.  84,  Ex.  6.) 

Let  u=  i    logs'mx.dx. 


Substitute  y  =  -  —  x. 


u  =  —  i    log  cos  y.dy  =  i    log  cosx.dx. 

2 
IT  n 

2  M  =       I    (log  sin  x  +  log  cos  x)  dx  =  |     log  (sin  x  cos  x)  dx 

s=  _ !!  log  (2)  +  (    log  sin  2a.(Za; 
2  Jo 

«=  - 1  log  (2)  +  ^  J"log  smx.dx. 

I    log  sinx.(/x  =  j  "log  sin. 7;. (?.»•+  j    logsinx.fia; 
=  it  -\-  I    log  sin.x\c?a;. 
Substitute  y  =  n  —  x. 


Phap.  VIII.J  DEFINITE  INTEGRALS.  103 

« 
I    log  sin  x.(fx  =  —  I    log  sin  y.d)/  =  i    log  sin x.dx  =  u. 

Hence  2?<  =  — ^  log(2)  +  M, 

and 

i<=  (    log  sin  a-. (/.r=  )    log  eosx.(7.f  =  —  -  los;(2).  [1] 

c/o  »/o      "  2     ■" 

(6)  f  e-''(f.r.  (v.  Art.  86  (a).) 

Let  u=  \    e  "- dx,     and  let  x  =  u y  ; 

then  ^l=  i   ae-'^-""- dij  =  \    ae-^'-^'cU*, 

?<  r"e-«=  r7«  =  u-  =  j '  Y  (*"ae-''+^=''^=  da") cZx,    by  [3],  Art.  89. 
But 

J)  2  1+  .t-2 

Hence  %e  =  -  I     — ^ — .;  =  -, 

2  Jo    1  +  .r-      4 

and  I    e'-dx—     Vtt.  [2] 

Jo  2 

rsinfflx^^^  rv.  Art.  HG(^).) 

^   '  J\i  X 

AVc  have  1  =  Ce'^da     if  .r  >  0.       (Art.  H2,  Kx.  6.) 

a:    Jo 


104 
Hence 


INTECJRAL   CALCULUS. 

]dx 
e~"  i,\nnix.(la]dx 


[Art.  92 


clx=  \       sinm.f  I    e-'^Ula 

ox  Ji)      \  J»  J 

Jo    \Jo 
=  i    fi    e'"s'mmx.(LAda,hy[3],Avt.H\). 

=  r4^.  (Art.  82,  Ex.  7.) 

Jo    a-  +  Vl' 


Therefore 


J**sinw 
0  X 


'.dx=      -     if  m>0 


=  - 1     if  m  <  0 
=      0     if  m  =  0 

EXAMFLKS. 


[3] 


by  Art.  82,  Ex. 


(1)    J    a;  log  sin  a;,  da;  =  _  ZL  iog(-2). 

Siifjyestion :  let  .x-  =  tan  i 


e  "'^'da; 


Jo  X 


2a 


.dx      =0     if  m<-  1     or     m>l 


=  -     if  til,  =  —  1     or     ?yi  =  1 


=  -     if   -1  <m<l. 


Chap.  VIII.]  DEFINITE    INTECUAI.S.  105 

(6)     I      — —dx=--  Suqqestioit  :   iiitt'<i;r:itv  l»v  parts. 

93.    Differentiation  or  integration  tvith  respect  to  a  fjumitifii 
^vhit•h  is  independent  of  x.  (v.  Art.  .s'J.) 

((()   We  have  )    e''Ulx  =  --  (Art.  .s2,  Ex.  C.) 

Dififereutiate  Ix^tli  nu'iul)ers  with  respect  to  a, 

I     (  —  xe  '"dx)  = ^,     or      I    xe~"'dx  =  — 

Differentiate  again, 

Cx^e  "'dx^^. 
Jo  a^ 

Differentiating  n  times, 

f  x^e  "Hlx  =  -^.  [1]      (v.  Art.  Kr,,  Ex.  3.) 

Jo  a""^^ 

(5)    We  have  Ce  '-'dx  =  1  ^.  (Art.  <»2,  Ex.  3.) 

c/o  2   a' 

Differentiating  n  times  with  respect  to  a, 

r'^n     „.^  7        1.3.5... (2n- 1)      1^  r„, 

(v.  Art.  8r.,  Ex.  2.) 
( r )    We  have  ie  "dx  =  ^ .  ( A rt .  .S2 ,  Ex .  6 . ) 

J(i  c 

Multiply  by  dc,  and  integrate  from  't  to  6, 

Hence    f' e  "-e  '•' ^^    =log-.  [8] 

Jo  a;  a 


106  INTEGRAL  CALCULUS.  [Abt.  94. 

(d)  r\-c?x=-4-. 

Jo  a  +  1 

Multiph"  by  f?a,  and  integrate  from  h  to  a, 

J)    \Jb  j  J/,    a  +  1 

Examples. 

(1)  From     f"'-*^  =  -—     obtain 
t/o   a-  +  «      2  ^/(t 

f"        tfx         ^ TT  1.3.5... (2?i-l)  ^       1 

.T"  (7a-  = obtain 

0  11+1 

f '  a;"  ( log  a;)"'  rfa-  =  (-!)"• — — :• 

Jo       ^    ^         .        ^  (n+l)"'+^ 

(3)  From     |     e~°''cosma;.c7.x-  =  -— ^^ — -     obtain 

^    ^  Jo  a-  +  7/r 

I     cos  mx.dx  =  ^log      — — -  )• 

» 'o  X  \a'  +  my 

(4)  From     |     e'"^ fi'mmx.dx  =  — ol)tain 

^  ^  Jo  a'  +  m^ 


X 


sin  mx.dx  =  tan  ' tan 


iK  m  m 


94.    Tbe  method  illustrated  in   Art.  93    can   be  applied  to 
miicli  more  complicated  forms. 

(a)  f  e  ''  ^' .  dx.  (v.  Art.  80,  Ex.  2.) 


CliAi'.   VIII.]  DEFINITE    INTEGIIALS.  JQj 

Let  21=  i    e  ''  ^-dx; 

da  v.. 


then  ^  =  _2r^*.e 


.i- 
Substitnte  z  =  '\ 


and  ~=—-2\    e  "  '■  dz  =  —  2  (    e  ^'  '-  dx  =  —  2u. 

da  Jo  J,) 


Hence  —  =  —  '2  da. 

u 

Integrating, 

log»  =  -2a+C, 

and  %i=C\e~^''. 

When   a  =  0,  u^  C  e  ""'dx  =  ^\Pir.     (Art.  92  (b)  [2].) 

Tlierefore  Ci  =  ^  Vtt, 

^""^dx^^-^.  [1] 

(6)  f  e-"'''cos/>.i-.rf.r.      (v.  Art.  8(5,  Ex.  2.) 

c/O 

Let  w=  I     e~"'-''  cosbx.dx^ 

then  Li*  =  _  I    a-e  ""'"  ^'mhx.dx. 

db         Jo 


Integrating  In'  i)arts, 


I     xe  "'"^'sinto.f/.r  =      -    I     e 

,/n  2  r/%/" 


-'''cos^.i;.»/.i-=  —u. 
'lor 


riieretore  —  = ;  «, 

db  2  a- 

dn  b     ,, 

—  =  --— rf&. 
w  2  a^ 


108  INTEGRAL   CALCULUS.  L-^T.  95. 

lutegraliug,  we  have 

4  a- 
or  xi=Cie  *"\ 

Wlien   &  =  0,         u  =  Ce  "'^'dx  =  '^^-         (Art.  92,  Ex.  3.) 

Jo  2  a 

Hence  u=  i    e  "''^ cos bx.dx  =  ^^e"*^.  [2\ 

Jo  2a 

Examples. 

(1)     I ..«.«  =  tan  '— . 

.'<)  X  a 

Uiggestion:  —- — -  =  2  ( 
1  +  ar         Jo 

95.    Introduction  of  ivuiginary  constants. 

C  COS  {af)dx.  (v.  Art.  86(c).) 

We  have  f  e  "'^''  dx  =  —  \^.  CArt.  92,  Ex.  3.) 

J.I  2  a 


JjCt  a^  =  c-  V  —  1  =  '•-  [  cos  -  +  \'—  1  sin  - 

Tiien  a  =  r/'cos''  -f-  V^  sin-")  =  ^--  ( 1  +  V^), 

(Art.  2.^.) 

and  L  =  ___l^^_=      1      (1_V=T). 

-^«      CV2(1  +  V-1)       2cV2 


2 
SdQoestion :  — ^ — -  =  2  I     oe  "'^'"^''''(/a. 


Chap.  VHI.]  DEFINITK    INTEGRALS.  109 

%Jo  2c  \ 2 

But  e  ''^^^^  '       =  cos  (r-'or')  -  V^  sin  (rar') . 

([;-i]Art.:n.) 
Therefore 

(    cos  (c-.r )  dx  —  V^  f  sin  (c" .r)  c7.f  =  J-  *  -  ( l  _  V^), 
./o  Jo  20  \2 

and 

f  cos  (c^a^)  dx  =  C  siu  (rx-)  dx  =  i-  J-.  [1] 

(Art.  17.) 
Let  c  =  1 , 

and  r  cos  (.r^)  c?.^;  =  C  siu  (.i-)  f/.c  =  ^  J^.  [2] 

If  we  substitute  y  =  .r  in  [2],  we  get 

r^:mdy  =r^,/,=j^.  [3] 

Jo     V2/  -^"      VZ/  ^l^ 

Gamma  Functions. 

96.  It  was  shown  in  Art.  84  that  I  [  log  -  ]  dx  is  finite  and 
determinate  for  all  values  of  7i  greater  than  —1,  and  inlinite 
when  n  is  equal  to  or  less  than  —1.  The  substitution  of 
y  =  log-  reduces  this  integral  to  |  y"e  'dy,  or,  what  is  the 
same  thing,  to  |  a;"e  'dx;  and  in  Art.  8G,  Ex.  3,  the  student 
has  been  required  to  show  that  this  integral  is  finite  and  deter- 
minate for  all  values  of  n  greater  than  —1. 

I  X"  e  ^dxz=z  —  X"  e  '  +  n  i  .<•"  '  e  ^dx, 
by  integration  by  parts. 


110  INTEGRAL   CALCULUS.  [Art.  9G. 

If  n  is  greater  than  zero, 

X"  e''  —  0     when  x  =  0, 

and  a;"e  '  is  indeterminate  when  x=  x.     Its  true  value  when 
a;=  00,  obtained  by  the  method  of  I.  Art.  141,  is,  however,  zero. 

Therefore  |    x"" e~'' dx  =  n  \     x"  ^e~'dx  [1] 

for  all  positive  values  of  n. 

If  n  is  an  integer,  a  repeated  use  of  [1]  gives 

Jx" e'" dx  =  nl  I     e'dx ; 

but  I     e~'rf.r      =  1, 

and  we  have  1     x"  e~'  dx  =  n !  [2] 

provided  that  n  is  a  ■pusitive  tvJiole  number. 

If   n   is  not  a  positive    integer,   but   is   greater   than    —  1, 
I     x"e~'dx  is  a  finite  and  determinate  function  of  n,  and  its 
value  can  be  computed  to  any  required  degree  of  accuracy  by 
methods  which  we  have  not  space  to  consider  here. 

I  af~^C'dx  is  generally  represented  by  r(n),  and  has  been 
very  carefully  studied  under  the  name  of  the  Gamma  Function. 
If  n  is  a  positive  integer,  we  have  from  [2] 

r(»  +  l)  =  n!.  [3] 

From  [3],  r(2)         =1.  [4] 

Since  r(l)         =(    x'^e'dx^Ce'dx, 

r(i)       =1.  [5] 

We  have  always  from  [1] 

r(n  +  l)=»r(70,  [6J 

if  n  is  greater  tlian  zero. 


Chap.  VIII.J 


DEFINITE   INTEGRALS. 


Ill 


Siuce  j     x"e  ^dx  is  infinite  when  n  is  equal  to  or  less  than 

—  1,  it  follows  from  the  definition  of  r(n)  thtit  r(>«)=  ^  'f 
n  is  equal  to  or  less  than  zero.  It  has,  however,  been  found 
convenient  to  adopt  formula  [6]  as  the  definition  of  r(7i)  when 
n  is  equal  to  or  less  than  zero,  and  to  restrict  the  original  defi- 
nition to  positive  values  of  n.  The  result  easily  deduced  is 
that  r(?i)  is  infinite  when  n  is  equal  to  zero  or  to  a  nt-gative 
integer,  but  is  finite  and  determinate  for  all  other  values  of  u. 


97.    We  may  regard  the  formula 

r(«  +  l)  =  «r(n) 

as  a  sort  of  reduction  formula;  and  since  each  time  we  applv 
it  we  can  raise  or  lower  the  value  of  n  by  unity,  we  can  obtain 
any  required  Gamma  Function  by  the  aid  of  a  table  containing 
the  values  of  T  (n)  corresponding  to  the  values  of  n  between 
any  two  arbitrarily  chosen  consecutive  whole  numbers. 

Such  tables  have  been  computed,  and  we  give  one  here  con- 
taining the  common  logarithms  of  the  values  of  T{n)  fi-om 
w  =  1  to  n  =  2.  Tire  table  is  carried  out  to  four  decimal 
places,  and  each  logarithm  is  printed  with  the  characteristic  9, 
which,  of  course,  is  ten  units  too  large,  the  true  characteristic 
b6ing  —1. 

io  +  iogrr(n). 


n 

0 

1 

. 

3 

* 

6 

6 

7 

8 

9 

1.0 

9975 

9951 

9928 

9905 

9883 

9862 

9841 

9821 

9802 

l.l 

9.9783 

9765 

9748 

9731 

9715 

9699 

9684 

9669 

%55 

9642 

1.2 

9.9629 

9617  i  9605 

9594  '  95S3 

9573 

9564 

9554 

9546 

9538 

1.3 

9.9530 

9523  1  9516 

9510  9.S05 

9500 

9495 

9491 

9487 

94S3 

1.4 

9.9481 

9478 

9476 

9475 

9473 

9473 

9472 

9473 

9473 

9474 

1.5 

9.9475 

9477 

9479 

9482 

9485 

W88 

9492 

94% 

9501 

9506 

1.6 

9.9511 

9517 

9523 

9529 

9536 

9543 

9550 

9558 

95r.6 

9575 

1.7 

9.9584 

9593 

9603 

9613 

9623 

9633 

9644 

%56 

9667 

%79 

1.8 

9.9691 

9704 

9717 

9730 

9743 

9757 

9771 

9786 

9800 

9815 

1.9 

9.9831 

9846 

9862 

9878  9895 

9912 

9929 

9946 

9964 

9982 

112  INTEGRAL  CALCULUS.  [Art.  97. 

Such  a  table  enables  us  to  compute  with  Gamma  Functions 
as  readily  as  with  Trigonometric  Functions,  and  consequently 
the  problem  of  obtaining  the  value  of  a  definite  integral  is 
practically  solved  if  the  integral  in  question  can  be  expressed 
in  terms  of  Gamma  Functions. 

For  example,  let  us  consider 

(a)  I    a.*"e  "■'dx. 

Let  y  =  ax ; 

then  I     x"e  "^dx  = |    y"'e~''dy  = |     x"e-'dx. 

Jo  a"''c/o  a"'^Vo 

Hence     rVe-°'c?a;  =  ^  ^^  +  ^\  [1] 

c/o  a"'*' 

provided  that  a  is  positive  and  m>  — 1. 

(6)  C  a™  Aog  \Xdx.  (v.  Art.  84,  Ex.  4.) 

•     Let  y  =  —  \ogx. 

then  rVAogiy'dx=  ('  y"e^'^^'^»dy. 

Hence,  by  [1], 

if  m>—l  and  n  >  —  1 . 

(c)  Ce-^'dx. 

Let  y  =  x^; 

then  \    e'''-dx=\\     ^dy  =  \\    x'^e'dx. 

v/o  Jo        Vy  »^" 

Hence      C  e"dx  =  ^r  (^) .  [8] 


Cu.vi'.  VIII.]  DEFINITE    INTKGIJALS.  113 

98.  r  x"'  1(1-  .*•)"   '  (/.«•  =  7i  (m,  n)  [1] 

is  au  exceedingly  important  integral  tluit  can  be  expressed  in 
terms  of  Gamma  Functions  ;  it  is  known  as  the  Beta  Fniicdon, 
or  the  First  Eulerian  Integral,  r{n)  being  sometimes  calU-d  tin- 
Seco7id  Eulerian  Integral. 

In  the  Beta  Fnnction,  m  and  n  are  positive,  and  B{m,u)   is 
always  finite  and  determinate.  (v.  Art.  b4,  Ex.  o.) 

In  (  x"'  ^  {I  —  x)"hlx     let     v^l-x-, 

and  we  get 

fx-^-'il-xy  'r/.r=  fV  '(1  -.'/)'"  ''/.V, 

or  B{m,n)  ^  B{n,ni).  [2] 

In  f  a-"'  1  ( I  -  x) "  1  (/.f      let      x  =  ^L^ , 

and  we  get 

(•'.'.-(1  -■>■)■  '.te=  C:!-:!*^^    =  f  ^^_,te. 

Jo  ^  Jo   (1 +?/)"•+»  Jo     (I  -|-.r)'"+" 

.    Hence  i    ^'-' clx  =  B{m,n).  [:?] 

Jo    (I  +a;)'"^"  *■  ■' 

We  have  seen  in   [1]  Art.  !»7  {a)   that 

c/o  a"*^' 

Hence  V{m)=  \     a'"x"*'^e"^dx, 

T(m)a"  'p  ••=  i     a-+"  ia-'"-'e-''<'+''rfx, 

r(m)  r"a"  'e  "r/a  =  T  .r'"  Y  f  a-^^-^e-^'+'da^lx, 

Jo  (l+x)"^" 


114  INTEGKAL    CALCULUS.  [AUT.  99. 

Therefore        ^i^il')  =  f       ^"" dx ;  [4] 

r(m  +  H)      Jo    (1  +0;)"'+"  ■-  -^ 

or  by  [3], 

^         ^     Jo  ^  ^  r(m4-w) 

If  H  =  l— m,   tlieu  since  r(l)=l, 

f '_J— _  dx  =  r"  ^^  (/.r  =T(m)T(l~  m) .  [61 

Jo  {i-xy        Jo    1+a;  y   ^    y         )  l  j 

Formula    [G]   leads   to   an  interesting  coulirmation  of   Art. 
92(5). 

Let  m  —  ^,  and  we  have  from  [(>] 

Substitute  2^  =  V-'"' 

and  we  have         I     ^ =  2  I     — -—  =  ir. 

Jo    (l+a-)V-f         Jo    1+2/' 

Hence  ra)=V'^;  [7] 

and  since  by  Art.  97  (c) 

jf%'V/.x-  =^r(^), 

99.    By   the   aid  of   formulas   [4],  [a],  and  [7]   of  Art.  98 
a  number  of  important  integrals  can  be  obtained. 
For  example,  let  us  consider 

I  "  sin".i'.'/.r,  where  n  is  gjicater  than  —  1. 

Let  //  =  siiix-, 


and 


bin" x.dx  =  I    ?/"(!—  y-y'Hly. 
0  «/o 


Chap.  VIII.]  DEFINITE   INTEUUALS.  115 

Let,  now,  ^  =  ff-, 

and 


But 


b(^^\  1^  =     V   I    )  by  [.>]  Art.  98. 


by  [7]  Art.  98. 


^i$ 

-) 

Hence 


C\ur. ,,.  =  -/'- ±J-4.  [I] 

Jo  2     j^fn   ,   i\ 


rf^  +  1 


If  n  is  a  whole  number,  this  will  reduce  to  the  result  given 
in  Art.  87,  Ex.  5. 

EXAMPLKS. 


(2)     I    sin".'ccos'"a;.c/u;  =  — ^^ — - — r — 


2rf:^  +  i 


IIG 


INTEGRAL   CALCULUS. 


[Akt.  99. 


(3)  /'-^ 


_v^ 


i) 


Vi  —  x" 


"  ^C>l) 


r(i'  +  i)r 


r  ;.  +  !  + 


(4)     rV(l-a;'')''(/.t=- - 


>/i  + 


m  +  1 


CiiAi'.   IX. J  LENGTHS    Oi''    CUKVES.  117 


CHAPTER    IX. 

LENGTHS    OF    CURVES. 

100.   If  we  use  lecttinguhir  eoordiuates,  we  have  seen  (I.  Art. 
27)  that 

tanr  =  g,  [1] 

and  (I.  Arts.  52  and  181)  that 

(Is-  =  (J,i-  +  dy\  [2] 

From  these  we  get  siur  =  -^i  [3] 


[4] 


COSr  =  — , 
(Is 

by  the  aid  of  a  little  elementarv  Trigonometry. 

These  formulas  are  of  great  importance  in  dealing  with  all 
properties  of  curves  that  concern  in  an}-  way  tlie  lengths  of  arcs. 

We  have  already  considered  the  use  of  [2]  in  tlie  first  volume 
of  the  Calculus,  and  we  have  worked  several  examples  by  its 
aid  in  rectification  of  curves.  Before  going  on  to  more  of  the 
same  sort  we  shall  find  it  worth  while  to  ol)tain  the  equations  of 
two  very  interesting  transcendental  curves,  the  catenari/  and  the 
tractrix. 

The  Catpnary. 

101.  The  common  aUpiutn/  is  the  curve  in  which  a  uniform 
heavy  flexible  string  hangs  when  its  ends  are  support<'d. 

As  tlie  string  is  flexible,  the  only  force  exerted  by  one  portion 
of  the  string  on  an  adjacent  portion  is  a  pull  along  the  string, 
which  we  shall  call  the  tension  of  tlie  string,  and  shall  n-present 
by  T.  T  of  course  has  different  values  at  dilfcrcnt  points  of  the 
string,  and  is  some  function  of  the  coordinates  of  tlu'  i)oiiit  in 
question. 


118 


INTEGRAL  V)ALCULUS. 


[Art.  101. 


The  tension  at  any  point  has  to  support  the  weight  of  the  por- 
tion of  the  string  Ik'Iow  tlie  point,  and  a  certain  amount  of  side 
pull,  due  to  the  fact  that  the  string  would  hang  verticalh'  were 
it  not  that  its  ends  are  forciljly  held  apart. 

Let  the  origin  be  taken  at  the  lowest  point  of  the  curve,  and 

suppose  the  string  fastened 
at  that  point. 

Let   s   be  the   arc   OP, 
P  being  any  point  of  the 
string.  As  the  string  is  uni- 
form, the  weight  of  OP  is 
proportional  to  its  length ; 
we  shall  call  this  weight  ms. 
This  weight   acts  verti- 
cally downward,  and  must  be  balanced  by  the  vertical  effect  of  T, 
which,  by  I.  Art.  112,  is  T^sinr. 

Hence  Ts\nT  =  ms.  (1) 

As  there  is  no  external  horizontal  force  acting,  the  horizontal 
effect  of  the  tension  at  one  end  of  any  portion  of  the  string  must 
be  the  same  as  the  horizontal  effect  at  the  other  end.  In  other 
words,  TcosT  =  c  (2) 

where  c  is  a  constant.     Dividing  (1)  by  (2)  we  get 


=  —  tan  T, 
'ni 


where  a  is  some  constant, 
tion  in  terms  of  x  and  y. 


From  this  we  want  to  get  an  equa- 


hence 

or 

and 


tanr  =  Vsec^T  —  1  =  \  -rr, 

-«'(r^•)• 

ads 


(a'  +  s')^ 


=  dx. 


Integrate  both  members. 


CiiAi>.  IX.]  LENGTHS    OF   CURVES.  HO 

a  \og{s  +  VoM^  .s~)  =  ./•  -I-  C ; 
when  a;=  0,  s=  0, 
hence  C=aloo[a, 

and  log  (s  4-  Vo-  +  .s")  =  '-  +  log  a, 

s  +  -\/a--{-  tr   =rte«, 
Va^+  «"  =  <^fca  —  s, 


s  =  -  (e«  —  e  «)  =  a  tan  r       by  (3) .  ; 

I 

Hence                                    a^=  -  (e^- e"!),  i 

and                                             ?/  =  ^  (('a  -f-  (?-!)  +  C.  1 

If  we  change  our  axes,  takhig  the  origin  :it  a  point  n  units  i 

below  the  lowest  point  of  the   curve,  y  =  a  when  x  =  0,  and  i 

therefore  (7=0,  and  we  get,  as  the  equation  of  the  catenary,  '• 

Example.  ; 

Find  the  curve  in  which  tiie  cables  of  a  suspension-bridge  ; 
must  hang.                          7>*-^7';„*^r             Aiis.    A  i)aralu)la. 

A^T^    T/»- r,™/,«.      ^^-,  „^^.^ 

102.   If  two  particles  are  attached  to  a  striilg,  and  rest  on  a-^*  * 
rough  horizontal  plane,  and  one.  starting  with  the  string  stretclied, 
moves  in  a  straigiit  line  at  right  angles  witii  tlie  initial  iiosition 

of  the  string,  dragging  tlie  other  particle  after  it,  the  patli  of  the  , 
second  particle  is  called  the  trartrix. 

Take  as  the  axis  of  X  the  path  of  the  fii-st  particle,  and  as  ^ 

the  axis  of  Y  the  initial  position  of  the  string,  and  let  a  be  ; 


120 


INTEGRAL    CALCULUS. 


[Akt.  102. 


the  length  of  the  strinof.     From  the  nature  of  the  curve  the 
string  is  always  a  tangent,  and  we  shall  have  for  any  point  P 

y 


=  —  SUIT, 
(( 

for  r  lying  in  the  fourth  (juadrant  has  a  negative  sine. 

Y 


[1] 


hence 


and 


cr  asr      dx^  +  dy^ 

y\d.r  +  dy-)  =  ahly\ 

yhlx-  =  {<r-y^)dy^, 

dx  =  ± ■ 


is  the  differential  equation  of  the  tractrix. 

On  the  right-hand  half  of  the  curve  t  is  in  the  fourth  quadrant, 

-^  or  tan  t  is  neijative,  and  we  shall  write  the  equation 
dx 


dx 


(^(C-  —  f-)kdy 


[2] 


If  we  allow  the  radical  to  be  ambiguous  in  sign  we  shall  get 
also  the  curve  tliat  would  be  described  if  the  first  particle  went 
to  the  left  instead  of  to  the  right.  The  tractrix  curve,  generally 
considered,  includes  these  two  jiortions. 

Integrating  both  members  of  [2],  and  determining  the  arbi- 
trary constant,  we  get 


X  =  —  \l  <r  —  y-  -\-  (I  lo<j 
as  the  equation  of  the  tmdrix. 


a  +  Va--  V 


[3] 


A' ■ 

CHAP.  IX.]  LENGTHS   OF   CURVES.  "      ^        121 

EXAMPLKS.  ^,^      C.-.  .^'.e^o^ 

S-    ^-^^a 

(1)  Show  In-  Art.   102  (1)    that    in  tlie  tractrix    s  =  ,i\^' 

if  .s  is  lueasured  from  the  starting-point.  ^ 

(2)  Find  the  evolute  of  the  tractrix.      (1.  Art.  9:3.) 

Rectification  of  Curves. 

103.  In  finding  the  length  of  an  arc  of  a  given  curve  we 
can  regard  it  as  the  limit  of  the  sura  of  the  differentials  of  the 
are,  and  express  it  by  a  definite  integral. 


We  shall  have 


=  C-sldx"  +  cbf. 


Of  course  in  using  this  formula  wc  must  express  V'/.*'-  -|-  f^.V* 
in  terms  of  x  only,  or  of  y  only,  or  of  some  single  variable  on 
which  X  and  ]j  depend,  before  we  can  integrate. 

For  example  ;  let  us  find  the  length  of  an  arc  of  the  circle 

ar  -f  r  =  «'• 

2.T.d.c  +  •>ii.(hi  —  0, 
,  x.dx 


=«I 


dx 


V«-  —  af 

dx 


f"     dx  ira 

The  length  of  a  (juadrant  =  a  I      =  —  ; 

,-.   the  length  of  a  circumference  =  2  7ra. 


122  INTEGRAL  CALCULUS.         [Art.  104. 

Length  of  Arc  of  Cycloid. 
104.    For  Hie  c^'cloid  we  have 

X  =  aO  —  a  sin  0  ] 

^y-  (I.  Art.  99.) 

?/  =  «    —  acos^J 

dx  =  (i{l  —  cosO)dO  =  yds, 


$  =  vers 


-i.V 


de=^  ''.'/  dy 


"  ^i-yy      y'      V2«y/-/ 


\    a      a^ 


dx  = 


?^dy 


V2  ay  —  y- 

CZ6-' = dx^ + df = ^^^^^  =  '^«^y\ 

2  ay  —  ?/-      2a  —  ?/ 


ds  =  V2  a 


dy 


V2  a  —  y 

s  =  V27(  r'^=^=  =  2V27t(V2;rr7o  -  V2;r:^) 

»/»o  V2«  —  ?/ 
If  the  arc  is  measured  from  the  cusp,  ?/„  =  0, 


s  =  4  a  -  2  V27i  V2  a  -  y,.  [1] 

If  the  arc  is  measured  to  the  highest  i)oiiit,  ?/i  =  2  a, 

.s  =  4a. 
The  whole  arch  =  8«. 

Example. 

Taking  the  origin  at  the  vertex,  and  taking  the  direction  down- 
ward as  the  positive  direction  for  y,  the  equations  become 

X  =  ((6  +  a  sin  6 


(I.  Art.  100.) 
y  =    a  —  ac      ■  ■ 

Show  that  s  =  2yj2tiy  when  the  arc  is  measured   from  the 
summit  of  the  curve. 


Chap.  IX.] 


LENGTHS    OF    CritVKS. 


123 


10.").    "We  can  rectify  the  cycloid  without  climiiiatiiij;  B. 
X  =  ad  —  a  sin  ( 
y  =  a  —  ocosi 

dx^  +  d;/-  =  2  a-de'i  1  -  t'os^) , 


and 


s  =  a^2  i  (1  -cos$)\d6. 
If  ^y  =  0  and  6i  =  2  tt,  we  get  .s  =  8a  as  the  whole  curve 


cos  - 
2 


106.    Let  ns  find  the  loif/lh  of  an  arch  of  the  ejifci/cloid. 

X  =  {a  +  h)  cos^  -  b  eos^-^^-±-^^  I 

(I.Art.lo:»[l].) 


b 


dx  = 
dy  = 

ds'  =  (a+by'dff 


(„  +  /,)  shi  6  +  {a  +  b)  sin'-^^l 
{a  +  b)  cos^  -  {a  +  b)  cos^^-±-^'^l 


dd. 
dO. 


r|2-  2  (c-os'^'^  cos^  -\-  -sin'-^'^  sin^^l 

s  =  (a-\-b)^/2  C^'fl-vosjOy'ie, 

4b(a  +b)r        a  ^  ^i  n~\  rn 

8=       ^     ^    q  eos  — ^o-cos  — 6,    ■  [1] 

a         |_        2^>  2o       I 

To  get   a  coinpU'te  arcli  we    must   h't  ^„=ii  and   ^,  =  :i- n-. 

«/ 
Hence,  for  a  whole  arch, 

Hb{a  +b) 


124  INTEGRAL   CALCULUS.  [Art.  107. 

Examples. 

(1)  Find  the  length  of  an  arch  of  a  hvpocycloid. 

'      /  8b(a-b) 

Ans.  s  =  . 

a 

(2)  Find  the  length  of  an  arch  of  tiie  curve    x^-\-!/^  =  a^^ 
and  show  that  it  agrees  with  tie  result  of  Ex.  1. 

(v.  I.  Art.  109,  Ex.  2.) 
107.    Let  us  attempt  to  find  the  length  of  an  arc  of  the 


behave                     ^  x  ./..  _^  2 ;/ r^^  ^ 
a-             h- 

b-x  , 
cry 

a*y^                        u^  —  XT 

a2-eV 

a^-a^ 

dx". 


where  e  is  the  eccentricity  of  the  ellipse. 
The  length  of  the  elliptic  quadrant  is 

These  integrals  cannot  he  olttained  directly,  but 
[a^  —  eV\^ 
can  be  expanded  by  the  liinoniial  Theorem,  and  the  fractions 
formed  by  dividing  the  terms  of  the  result  by  \_a^  —  x-'\^  can  be 
integrated  separately,  and  we  shall  have  the  required  length 
expressed  by  a  series. 

A  more  convenient  way  of  dealing  with  the  problem  is  to  use 

an  auxiliary  angle.    Instead  of  \,  4"  '.^  =  1  we  can  use  the  pair 

of  equations 

X  =  a  sin<f>  .-r     .        .  ~^ 

,  ^  (I.  Art.  150), 

y  =  6C0S<^J  ^  ^' 


Chap.  IX. ]  LENGTHS   OK   CURVES.  12.') 

dx  =  a  cos(f>.d(f}, 
dy  =  —  b  sin  (f>.d(f>, 

ds^  =  (a'-cos-<^  -f-  bH\n-<f>)d<t>-  =  [a^  -  (a-  -  b^^Hm-tfyldffr 
=  a-fl  —  ^^~^"sin-<^') f?<^-  =  a-(l  -  e-.siir-»</<^', 
where  e  is  the  eccentricity  of  the  ellipse. 

s=a        (1 -e-sin-»^cZ<^  [3] 

•/fro 

=  a  f  '[l-^e-sin-</)-i-|'^"'^^i"'<^-i-i-ile''siii*"'<^-"]^'<^- 

•/fro 

For  the  arc  of  a  quadrant  wi'  have 
Sg  =  a\     \_\  —e~&\ir<i>y-d<j>.  [4] 

Example. 

(1)   Obtain  s,  as  a  series  from  [2],  and  also  fVoin  [4],  and 
compare  the  results  with  Art.  1>1,  Ex. ."). 

Polar  Formulce. 
108.    If  we  use  polar  coordinates  we  iiave 


ds  =  \l~d,^  +  r-  d4-,     ( I .  A rt.  -20 7 ,  Ex .  2 . ) 

tanc  =  ^,  (I.  Art.  207.) 

dr 

From  these  we  get,  l)y  Trigonometry, 

sine=^-^,  cos€  =  — . 

ds  (i-'i 

109.    Let  us  find  the  equation  of  the  curve  which  crosses  all 

its  radii  vectores  at  the  same  angle.     Here 

rdd)  adr       ., 

taue  =  a,     a  constant,  --^  =  a,  —-  =  a<p, 

dr  ^ 


126  INTEGRAL   CALCULUS.  [Akt.  110. 

a  log  ?'  =  <^  -+-  C,  r  =  e"    "  =  e"  e" , 

r  =  be':  (1) 

where  b  is  some  constaut  depeudiug  upon  tlie  position  of  the  origin. 
This  curve  is  known  as  tlie  Logarithmic  or  Equiangular  Spiral. 

110.  To  rectify  tlie  Logarithmic  *SpimZ.      We  have,   from 
109  (1), 

a  b 

'>' 
rd4>  =  adr,    ' 
ds^^dr  +  ?"(?(^2  =  (1  +  a^^di^ ; 

s  =  C{\  +  (r') i (?/•  =  ( 1  +  c(-) h(r,  -  r,) . 

Examples. 

(1)  Find  the  length  of  an  arc  of  the  parabola  from  its  polar 

equation 

r= 

1  -f-  cos  4> 

(2)  Find  the  length  of  an  arc  of  the  Spiral  of  Archimedes 

r  =  Uff). 

111.  To  rectify  the  Ca/rZ/oiYZe.     We  have 

r=2a(l-cos<^),  (I.  Art.  109,  Ex.  1), 

dr=  '2as'u\(f).d<f), 

d.s-  =  -1  a-iiin-(j>.d(j)-  +  I «-{]  —  coH<f>)-dcf>^ 
=  8  ct-d(f)-{  1  —  cos  <^) , 

s  =  2  ^2  .  a  ^i\  •-cos<^)V/<^  =  8  afcos^  -cos|n 

=  lOff  for  the  whole  perimeter. 


Chap.  IX.]  LENGTHS   OK   CriiVES. 


Involutes. 

112.  If  we  can  expi-ess  the  leugth  of  the  arc  of  a  <;ivon  curve, 
measured  from  a  fixed  point,  in  terms  of  the  eourdimites  of  it.s 
variable  extremity,  we  can  lind  the  equation  of  the  involute  of 
the  curve. 

We  have  found  the  equations  of  the  evohite  of  y=fj:  in  the 
form 

x'  =  x  —  p  cos  V 

y'  =  y-p  sin  V 


(I.  Art.  1)1). 


We  have  proved  that      tani'  =  tanr',  (I.  Art.  'J.j), 

fit' 
and  that  3^=1»  (I.  Art.  UG)  ; 

dp 


smr  =  -^, 
els' 

,     dx' 
cos  T  =  -— 

ds' . 


(Art.  100). 


Since         tanv  =  tanT',    v  =  t'   or    r=l.sO°  +  T'. 
As  normal  and  radius  of  curvature  have  opposite  directions, 
we  shall  consider  v  =  180°  +  r'. 

Then  sin  i'  =  —  sin  r'     and     cos  r  =  —  cos  t'. 

Hence  x'  =  x-\-  p-j-i  0) 

ds 

Since  dp  =  ds\ 

P  =  s'  +  l  (3) 

where /is  an  arliitr.irv  ((.nstmit.  Since  ./•  and // are  the  coordinators 
of  any  point  of  the  mvohtte,  it  is  only  necessary  to  eliminat*.'  x', 
y',  and  p  by  combining  equations  (1) .  (2),  and  (."5)  with  the  equa- 
tion of  the  e volute. 

As  we  are  supposed  to  start  with  the  e(iuation  <>f  the  evolute 
and  work  towards  the  equation  of  the  involute,  it  will  be  more 
natural  to  accent  the  letters  belonging  to  the  latter  curve  instead 


128  INTEGRAL   CALCULUS.  [Art.  112. 

of  those  going  with  tlic   former ;    and  our  eqnations  may  be 
written 

X  =  x'+p''^;  y  =  y'+p''k;  p'=S  +  L  (4) 

as  as 

Since  p'=l  when  .s  =  0,  it  follows  that  I  is  the  free  portion 
of  the  string  with  which  we  start.  (I.  Art.  97.)  By  varying  I 
we  may  get  different  involutes  of  the  same  curve. 

To  test  our  method,  let  us  find  the  involute  of  the  curve 

for  which  /  =  m.     We  must  first  find  s. 
2ychj  =  -^{x-m)-dx, 

9  m        y 

,  .,       2  X  -\-  ill   ,   o 

(Is-  = —  dx-, 

"dm 


■  , 1    V 

V3  m^m 

;}  V  3  m 

<-^ 

p  =  .s-  -\-  m  =  - 

;  V3  m 

-'-n 

m 

2  /  m 

{•2x  +  m){x-m.Y 

y 

,       X  —  ni. 

X-       .J      . 

.T  =  3.r'-f-m, 

V 

Substituting  in  (.^>)  tl 

le  values  of  x  and  y  just  obtained, 

we 

have 

y^  =  2  ?».»•' 

the 

equations  of  the 

rcfiuwed  involute. 

Ciivi'    IX.]  LENCTHS    OF   CURVES.  1 'il* 

Example. 
Find  an  involute  of  aif  =  x^. 

U.S.    An  involute  of  the  cycloid  is  easily  found.     Take  ecjua- 
lionsl.  Art.  100(C). 

X  =    aO  +  a  sin  0  | 
y  =  —a-\-a  cos  6  ) 
Let  p'  =  s, 

dx  =  a  ( 1  +  cos 6)de       =2  a  cos- ^ cW, 

6       9 
dii  =  —  a  sinO.iW  =—  2  a  sin  -  cos  -  (W, 

J  2        2 

els'  =  2  o2  dd-  ( 1  +  cos  6)  =  A  a-  dO-  QoA 

s  =  2aj  cos-(/^         =4asni^^' 

6       6 
X  =  x'  +  4  a  sin  -cos  -  =  x'  -\--2a  sin  9, 

Q 

y  =  y'  —  ia  sin--         =  y'  —  2  (<  ( 1  —  cos ^) , 

x'  =  a^  —  (I  sin  < 
y'  =    a  —  (( cos  < 

a  cycloid  with  its  cusp  at  the  summit  of  the  given  cycloid. 

Example.       "^ 
From  the  equations  of  a  circle 

x  =  a  cos  < 
y  =  a  sin  < 
obtain  the  equations  of  the  involute  of  the  circle.     Let  /  =  (» 
Ans.    x'=  a(cos0  +  0  sin  <^) 
y'=  a  (sin  «/>  —  </»  cos  <f>) 


130  INTEGRAL  CALCULUS.  [AuT.  114. 

Intrinsic  Equation  of  a  Curve. 

114.  An  equation  connecting  tlie  length  of  the  arc,  measured 
from  a  fixed  point  of  any  curve  to  a  variable  point,  with  the 
angle  between  the  tangent  at  the  fixed  point  and  the  tangent  at 
the  variable  point,  is  the  inti'insic  eqiiation  of  the  curve.  If  the 
fixed  point  is  the  origin  and  the  fixed  tangent  the  axis  of  X,  the 
variables  in  the  intrinsic  equation  are  s  and  t. 

We  have  already  such  an  equation  for  the  catenary 

s=atanT,  Art.  101  (3),     [1] 

the  origin  being  the  lowest  point  of  tlie  curve. 
The  intrinsic  equation  of  a  circle  is  obviously 

s  =  ar,  [2] 

whatever  origin  we  may  take. 

The  intrinsic  equation  of  the  tractrix  is  easil}'  obtained.    Vie 

have 

2/  =  -asinT,  Art.  102  (1), 

and  s  =  a  log- ;  Art.  102,  Ex.  1. 

y 

hence  s  =  alog(— csct) 

where  t  is  measured  from  the  axis  of  X,  and  .9  is  measured  from 

the  point  where  the  curve  crosses  the  axis  of  Y.    As  the  curve  is 

tangent  to  the  axis  of  Y,  we  must  replace  t  by  t  —  90°,  and  we 

get 

s  =  alogsccT  [3] 

as  the  intrinsic  equation  of  the  tractrix. 

Example. 

Show  that  the  intrinsic  equation  of  an  inverted  cycloid,  when 
the  vertex  is  origin,  is 

s  =  4asinT;  (1) 

when  the  cusp  is  origin,  is 

.s  =  4o(l  —  cost).  (2) 


CiiAi'.   IX.]  LENGTHS   OF   CI  HVKS.  l;jl 

115.    To  find  the  intrinsic  equation  of  the  epicveloiil  we  can 
use  the  results  obtained  in  Art.  lOG. 

di,==(a-\-b)feosO-i'os^-^^e\w=2(a+b)sm'^^^$Hm  — 6.(10, 
\  ^>      J  '2  b  2  b 

by  the  formulas  of  Trigonometry 

sin  a  —  sin^  =  2  cos  ^ (a  +  /?)  sin^(a  —  ^), 
cos/S—  cos  a  =  •2sin^(u  +  (3)  sin  ^(u  —  /3)  ; 

tan  T  =  —  =  tan  — ■ 6, 

dx  2  b       ' 

,  a  +  2b^ 

hence  t  =  — — —  0  ; 

2o 

^    ^    ^ '  1  —  cos 


i^^)byArt.ir)r,[i; 


therefore  s  =  ^-^^^^^±^H \  -  cos -^  r)  [1] 

a          \  a  +  2b  J 

is  the  intrinsic  equation  of  the  epicycloid,  with  the  cusp  as  origin. 
If  we  take  the  origin  at  a  vertex  instead  of  at  a  cusp 


^_^(a  +  2b) 
2  a 

+  t' 

' 

„,      4h(a+?>) 

sin- 
a 

a 

a 

+  2 

b 

„_ib(n+b) 

sin- 

a 

—  T 

or  s  =  — - — ^i— ^sin — -T  I  21 

(I  ((  +  2b 


is 


the  intrinsic  e(iuati«)n  of  an  epicycloid  ri-ferred  t<j  a  vertex. 


132  INTEGRAL  CALCULUS.         [Art.  116. 

Example. 
Obtain  the  intrinsic  equation  of  the  hypocycloid  in  the  forms 
4b(a-b) 


a  —  2b 


(1) 


4b(a  —  b).        a  ,_, 

.s= — ^ ^sni —  T.  (2) 

a  a  — 2  b 


116.   The  intrinsic  equation  of  the  Logarithmic  Spiral  is  found 
without  ditticulty. 

We  have  r=be'',  (Art.  109), 


and  s  =  Vl+a-(ri-ro).  (Art.  110). 

If  we  measure  the  arc  from  the  point  where  the  spiral  crosses 
the  initial  line,  Tq  =  b,  and  we  have 

s  =  b\/Y^a-{e^-\). 

In  polar  coordinates  t  =  <^  +  e,  and  in  this  case  €  =  tan~*a  ;  if 
we  measure  our  angle  from  the  tangent  at  the  beginning  of  the 
arc  we  must  subtract  c  from  the  value  just  given,  and  we  have 

s  =  Z;(VH-a-)(e"-l)  ; 

or,  more  briefly,        s  =  k{c^  —  1 ) ,  A:  and  c  being  constants. 

117.  If  we  wish  to  get  the  intrinsic  equation  of  a  curve  directly 
from  the  equation  in  rectangular  coordinates,  tlie  following  method 
will  serve  : 

Let  the  axis  of  X  be  tangent  to  the  curve  at  the  point  we  take 
as  origin.  r^  f  m^-^ 

tanr  =  '^';      le^Hof^  (1) 

and  as  the  ociuation  of  the  curve  enables  us  to  express  y  in  terms 
of  x,  (1)  will  give  us  x  in  terms  of  t,  say  x*  =  Ft  ; 


Ciiw.  IX.]  LENGTHS   OF    CUKVES.  133 

then  dx=F'T.dT,     <?^k /^:^P^^^  i\\yu]v  hy  ,ls:  \ 

c]^  =  /.V 1^    hut     ^-  =  cos  r  ;    ^  .  >^  "^yr"  ^ 

ds       ds        ds         ^       ^r  ^ 

hence  ds  =  sec tF't.^/t.     a/ S  ^  f-t^-^"^  ^-^^A'-) 

Integrating  both  nienilx'is  we  shall  have  the  reqiured  intrinsic 
equation. 

For  example,  let  us  take  x^  =  2  my,  which  is  tangent  to  the 
axis  of  X  at  the  origin. 

'^^  •Ixdx^'lmd]!,  '*'' 

a^  

«      ^J  ((X=  VlSQC''T.ClT, 


^    .    a>-^^^^ 


= 

tan  7  = 

X 

m 

r^.r 

= 

VI  SCi/r.dT, 

dx 
ds 

= 

COSt  = 

m  sec^ 

r  — 
(7.S 

ds 

= 

?»  secV.f?r, 

*--  n^ 


CX^  ds  =msQi-'^T.dr,  (1) 

=  0;         .-.   C'=0; 


g^^^^^^-^-f^^ 


pf^  ^  Examples. 


(1)  Devise  a  method  when  the  curve  is  tangent  to  the  axis 
of  I",  and  apply  it  to  ?/*=  2  ma;. 

(2)  Ohtain  the  intrinsic  equation  of  v'-'  = (x  —  wY- 

'It  in 

(3)  Obtain  the  intrinsic  equation  of  the  involute  of  a  circle. 
(Art.  113,  Ex.) 


134 


INTEGRAL   C^VLCULUS. 


[Art.  118. 


118.    The  evolute  or  the  involute  of  a  curve  is  easily  fouucl 
from  its  inti-insic  equation. 


\ 

si 

i 

r'N 

\(2) 

p. 

J--^/t 

If  the  curvature  of  the  given  curve  decreases  as  we  pass  along 
tlie  curve,  p  increases,  and 

s'  =  p-p^.  (I.  Art.  9G). 

If  the  curvature  increases,  p  decreases,  and 


Hence  ahva^'i 


,s  =p„  —  p. 

s'  =  ±{p  —  po); 


[1] 

p  =  — ,  (I.  Arts.  86  and  90). 

(It 


We  see  from  the  fi<i;ui-e  tliat  r'  =  r. 
I  lence 


-[^1.-(SIJ 


or,  as  we  shall  write  it  for  brevit}' 


(Is 


[2] 


119.    The  evolute  of  the  tmdrix  s  =  a  log  seer  is 

g  ^  ,j (Hog sec tI  '  ^  ^^  ^,^j^  ^^     ^j^^  catenary. 

The  evolute  of  the  circle  .s  =  ar  is 

s  =  (t'-H  =0,     a  point. 
Mm 


Chap.  IX.]                     LENGTHS   OF   CUKVES.  13;-, 

The  evolute  of  the  cycloid  s  =  4  a ( 1  —  cos r )  is  j:/t  //^/ nifc  £jti 

.  /i(i-cosT)r   <    •  ^^  /^^^t  sf. 

an  equal  cycloid,  with  its  vertex  at  the  origin.  J"  -«a  ZSU—  t  C^* 


Example; 


s-HicT-zh  ^^r 


(1)  Prove  that  the  evolute  of  the   logarithmic  spiral  is  an 
equal  logarithmic  spiral. 

(2)  Find  the  evolute  of  a  parabola. 

(3)  Find  the  evolute  of  the  catenary-. 

120.   The  evolute  of  an  epic3'cloi(l  is  a  similar  epicycloid,  with 
each  vertex  at  a  cusp  of  the  given  curve. 
Take  the  equation 

s  =  iM^^^l^A_eos-^A'Art.llo[l]. 
For  the  evolute, 

d[\  —  C(JS  '- T 

_46_(a_-M0      V  ((  +  2  }> 

a  ih 

■ih  ((I  -\-b)    .•        a  r,T 

s  = ^^ — — -sni r.  ri| 

a  +  2 1>  (i  +  -2b  "-  -■ 

The  form  of  [1]  is  that  of  an  epicycloid  referred  to  a  vertex 

as  origin  ;   let  us  lind  <i'  and  />',  the  radii  of  the  lixed  and  rolling 

circles. 

4b'(a'  +  b')    .         a'  ,      »   ,   , ,  -  r.>i  . 

s  = ^^ ■ ^sni -T,        l)v  Alt.  1 1.»  [2]  ; 

,       ,    hence,  4  6V +  //)  ^  4/.  (. +  /.)^         ^-^Jj^^jLllI 


dC-h 


•v; 


r/'  a  i^^H^ 


^  K 


o 


136  INTEGRAL   CALCULUS.  [AuT.  )2L 

Solving  these  equations,  we  get 


a  +  2b' 

h' 

ab 

a +  26 

a' 
h' 

a 
~b' 

and  the  radii  of  the  fixed  and  rolling  circles  have  the  same  ratio 
in  the  evolute  as  in  the  original  epic3-cloid ;  therefore  the  two 
curves  are  similar. 

Example. 

Show  that  the  evolute  of  a  Inpoc^'cloid  is  a  similar  hypo- 
cycloid. 

121.  We  have  seen  that  in  involute  and  evolute  r  has  the  same 
value  ;  that  is,  t  —  t'. 

If  s'  and  t'  refer  to  the  evolute,  and  s  and  t  to  the  involute^  we 
have  found  that 

8'  =  —!'', 

or  s'  =  — ^  —  /,      /  being  a  constant, 

cIt 

the  length  of  the  radius  of  curvature  at  the  origin. 
{s'  +  l)(W  =  ds, 

is  the  equation  of  the  involute. 

The  involute  of  the  catenary  s  =  a  taur  is,  when  /  =  0, 


<(  I  tun  T.(ir  =  a 


log  sec  T,     the  tractrix. 


Chap.  IX.]  LENGTHS   OF   CUllVES.  137 

The  involute  of  the  cycloid  .s  =  1  d  siiir  wlieii  /  =  0  i.s 

s  =  4a  i  smT.dr  =  4(i{\  —  cost), 

an  equal  C3'eloid  referred  to  its  cusp  as  origin. 

The  involute  of  a  cycloid  referred  to  its  cusp  .•<=  4*^(1  —cost) 
when  /  =  0  is 

.s  =  4a  I   (1  —  cosT)f?r  =  -la(T  —  siuT), 

a  curve  we  have  not  studied. 

The  involute  of  a  circle  .s  =  ar  when  I  =  0  is 


=«i 


122.   While  any  given  curve  has  but  one  evolutc.  it  has  an 
infinite  number  of  involutes,  since  the  equation  of  tlic  involute 


-^■=X\s  +  n<h 


contains  an  arbitrary  constant  / ;  and  the  nature  of  the  iu\olute 
will  in  general  be  different  for  different  values  of  /. 

If  we  form  the  involute  of  a  given  curve,  taking  a  particular 
value  for  /,  and  form  the  involute  of  this  involute,  taking  tlie  same 
value  of  ?,  and  so  on  indefinitely,  the  curves  ol)tained  will  con- 
tinually approach  the  logarithmic  spiral. 

Let  .s=/r  (1) 

be  the  given  curve. 


Jo  »/o 


s  = 
is  the  first  involute  ; 

.s=  f  (/  +  /r4-  C'fT.(lT)>h  =  !-  +  —  +  f    f'./V.fM 

i/ii  .'0  "  2       »'"  .'" 

is  the  second  involute  ; 

.  =  /T  +  ^  +  ^^  + +'i:+f\r,lr-^  (2) 

2       .3 !  u  I     .  '0 

is  the  nth  involute. 


138  INTEGRAL   CALCULUS.  [Art.  123. 

By  Maclamiii's  Theorem, 

But  s  =  0  when  t  =  0  ;  hence  fo  =  0,  and 

./V  =  .l,r     +i,=  +  £!|..+ 


o  !  4  ! 


as  n  increases  indefinite!}'  all  tlie  terms  of  (3)  approacli  zero 
(I.  Art.  133) ,  and  tlic  limiting  form  of  (2)  is 

s  =  ir  +  ^r  +  ^  + 

2  !       3  ! 

=  ?(^1+-  +  — +  — + -1 

V        1       2!      3! 

s  =  l{e'-l)  by  I.  Art.  133  [2], 

which  is  a  logarithmic  spiral. 

123.   The  equation  of  a  curve  in  rectangular  coordinates  ib 
readil}-  obtained  from  the  intrinsic  equation. 
Given  «=./V, 

we  know  that  sin  t  =  — , 

(Is 

and  cos  T  =  —  ; 

(Is 
heuce  dx  =  cos  rds  =  cos  t/V.^t, 

dy  =  sin  rds  =  sin  rfW.dT^ 

a;  =  I  cos  t/'tAt  \ 


y:=p,Urrr..lr\ 


Chap.  IX  ]  LENGTHS   OK   CURVES.  139 

The  elimination  of  t  I)i'twecn  these  equations  will  give  us  tiic 
equation  of  the  curve  in  terms  of  x  ami  y.  Let  us  apply  this 
method  to  the  catenary. 

s  =  « tanr, 

ds  =  a  sec^T.dr, 


X  =  a  fseer.f?r  =  a  log  x'^^ti^ULT, 
Jo  \  1  —  sinr 

y  =  (I  i  sec  T  tan  r.dr  =  a  (sec  t  —  H , 


1  +  sinr 
1  —  sinr' 

2z  X 

e  "  —  1       e"  —  e' 


e  «  + 1      f^«  -f-  e  a 

the  equation  of  the  catenar}-  referred  to  its  lowest  point  as  origin. 

Curves  in  Space. 

124.  The  length  of  the  arc  of  a  curve  of  douhle  eiu-vature  is 
the  limit  of  the  sum  of  the  chords  of  smaller  ares  into  which  the 
given  arc  maybe  broken  up,  as  the  number  of  these  smaller  ares 
is  indefinitely  increased.  Let  (x,y,  z) ,  (x  +  d.v,  >/  -}-  A//.  2  -}-  Az) 
be  the  coordinates  of  the  extremities  of  any  one  of  the  small  ares 
in  question;  c?a;,A//, Az  are  infinitesimal;  Vr/af^-|-A//-'-h Az^  is  the 
length  of  the  chord  of  the  arc.  In  dealing  with  the  limit  of  the 
sum  of  these  chords,  any  one  may  be  replaced  by  a  (piantity  dif- 
fering from  it  by  infinitesimals  of  higher  order  than  the  first. 
y/dx-  +  dy-+  dz'  is  such  a  value  ; 


hence 


140  INTEGRAL   CALCULUS.  [Art.  124. 

Let  us  rectify  the  helix. 

X  =  a  cos  6  1 

^J  =  asinO  L  (1.  Art.  214.) 

z  =  kd         J 
dx  =  —  asinO.dO, 
dy  =  a  cos6.dd, 
dz  =  kde, 
ds^  =  {a^  +  Jr)d6\ 
s  =  {a-  +  A-) i  (  "\W  =  \/a^+k\Oi  - 0,) . 

Examples. 

(1)  Find  the  length  of  the  curve  (y  =  — ,  z  =  ^\ 

V        -la  Gay 

Ans.    s  =  x-j-z-\-l. 

(2)  y  =  2  \/ax  -  a;,  2  =  a;  -  |  J-.       Ans.  s  =  x  +  y-z-\-l. 


CriAi-.  X.] 


141 


CHAPTER   X. 


125.  We  have  found  and  used  a  formula  for  the  area  bounded 
by  a  given  curve,  the  axis  of  X,  and  a  pair  of  ordiuates. 


=fydx. 


We  can  readily  get  this  formula  as  a  definite  integral 
area  in  the  figure  is  the  sum  of  the 
slices  into  which  it  is  divided  by  the 
ordinates ;  if  ^x,  the  base  of  each 
slice,  is  indefinitely  decreased,  the 
slice  is  infinitesimal.  The  area  of 
any  slice  differs  from  y\x  by  less 
-than  A?/Ax,  which  is  of  the  second 
order  if  Ax  is  the  principal  infini- 
tesimal.    We  have  then 


r. 


Mie 


.         limit    ^,"^1    ^ 

^  =  ^x=o  ^  y^"" 


Hence 


-=X' 


ydx. 


bv  I.  Art.  ir.l, 


[1] 


If  the  curve  in  question  lies  above  the  axis  of  X,  and  r^  is 
less  than  a^i,  each  ordinate  is  positive,  each  Ax  is  positive,  each 
term  of  the  sum  whose  limit  is  required  is  positive,  the  .sum  is 
positive,  and  the  limit  of  the  sum  or  the  area  sought  is  positive. 
If,  however,  the  curve  lies  below  the  axis  of  A',  and  .r„  is  less 
than  x,,  each  ordinate  is  negative,  each  Ax  is  positive,  each 
term  of  the  sum  is  negative,  the  sum  is  negative,  and  the  limit 


142  LNTEGiiAL  CALCULUS.  [Alir.  125. 

of  the  sum  or  the  area  sought  is  negative  If,  then,  the  curve 
happens  to  cross  tiie  axis  between  .i-y  and  x'l,  Ibruuila  [Ij  gives 
us  the  difference  between  the  portion  of  the  area  above  the  axis 
of  X  and  the  portion  below  the  axis  of  X,  but  throws  no  light 
upon  the  magnitudes  of  the  separate  portions.  Consequently, 
in  any  actual  geometrical  problem  it  is  usually  necessary  to  find 
the  portion  of  the  required  area  above  the  axis  of  X  and  the 
portion  below  the  axis  of  X  separately  ;  and  for  this  purpose 
it  is  essential  to  know  at  what  points  the  curve  crosses  the  axis. 
Indeed,  if  the  problem  is  in  the  least  complicated,  it  is  neces- 
sary to  begin  by  carefully  tracing  the  given  curve  from  its 
equation,  and  then  to  keep  its  form  and  position  in  miud  during 
the  whole  process  of  solution. 


Examples. 

(1)  Show  that  I    \'ch/  is  the  area  bounded  by  a  curve,  the 

axis  of    Y,  and    perpendiculars   let  fall   from  the  ends  of  the 
bounding  arc  upon  the  axis  of  Y. 

(2)  If  the  axes  are  inclined  at  the  angle  w,  show  that  these 

formulas  become 

\ 


A  =  sin  w  I    ydx  =  sin  w  |    xdy. 


(3)  Find  the  area  bounded  by  the  axis  of  X,  the  curve 
ar^  +  4_v  =  0,  and  the  ordinate  of  the  point  corresponding  to  the 
abscissa  4.  Ans.  b\. 

(4)  Find  the  area  bounded  by  the  axis  of  X,  the  curve 
2/ =  ar',  and  the  ordinates  corresponding  to  the  abscissae  —2 
and  2.  Ans.   8. 

(5)  Find  the  area  bounded  by  the  axis  of  X,  the  axis  of  Y, 
the  curve  .v  =  cosa;,  and  the  ordinate  corresponding  to  the 
abscissa  Stt.  Ans.  6. 


Chap.  X.] 


AUKAS. 


U: 


120.    lu  polar  foordi nates  wc  cixn  regard  the  area  1)etween  two 
radii  vectores  and  the  curve  as  the  Hinit  of  the  sum  of  sectors. 

The  area  in  question  is  tlie  sum 
of  the  smaHer  sectorial  areas,  any 
one  of  which  differs  from  ^  >-^A<^  In- 
less  than  tlie  ditlerence  between  the 
two  circular  sectors  ^(r  +  A;-)'''A<^ 
and  ^rA<^;  that  is,  by  less  than 
(A/-)^'A(/> 
2 


/•A;-A</)  + 


,  which  is  of  the 


second  order  if  A<^  is  the  [)rincipal  infinitesimal. 
Hence  A  =  ^^^"^^^[i^lV'A  J , 

A  =  ^i    rdcf>. 


127.    Let  us  find  the  area  between  the  catenary,  the  axis  of 
X,  the  axis  of  Y,  and  any  ordinate. 


A  =  Cydx  =-  i(e^ 


-\-e'a)(lx, 


but 
Hence 


A  =  ^{ea-e-a), 


(C«  —  €")  = 


by  Art.  101. 


A  =  a.5, 


and  the  area  in  question  is  the  length  of  the  arc  multiplied  by  tlu- 
distance  of  the  lowest  point  of  the  curve  from  the  origin. 


128.    Let  us  find  the  area  l)etween  the  ti-actrix  and  the  axis 
ofX. 


We  have 


(^!/. 


A  =  Cydv  =  -  \(hj\l(r  -  if. 


(.\rt.  102.) 


144  INTEGHAL   CALCULUS.  [Art.  129. 

The  area  in  question  is 


.'..  4 

iiicli  is  the  area  of  tlie  (juadrant  of  a  circle  with  a  as  radius. 


Example. 

Give,  by  the  aid  of  infinitesimals,  a  geometric  proof  of  the 
result  just  obtained  for  the  tractrix. 

121).  In  tlie  last  section  we  found  the  area  between  a  curve 
and  its  asymptote,  and  obtained  a  finite  result.  Of  course  this 
means  that,  as  our  second  l)ounding  ordinate  recedes  from  the 
origin,  the  area  in  question,  instead  of  increasing  indefinftelj, 
approaches  a  finite  limit,  which  is  the  area  obtained.  "Whether 
the  area  between  a  curve  and  its  asymptote  is  finite  or  infinite 
will  depend  upon  the  nature  of  the  curve. 

Let  us  find  the  area  between  an  liyi)erbola  and  its  asymptote. 

The  equation  of  the  h}  [)erbola  referred  to  its  asymptotes  as 
axes  is  -'  4_  /  2 


Let  (D  be  tiie  angle  l)etween  the  asymptotes  ;  then 

A  =  sin  (J)  I  yax  = sm  (»  I     —  =  ao 

.yii  ■  4  J(i      X 

Take  the  curve  y'x  =  -i<r{2a  —  x) , 

2  _  I    2   2  (I  —  .^' . 


any  value  of  x  will  give  two  values  of  }/  e(jual  with  opposite 
signs  ;  tlierefore  tlie  axis  of  x  is  an  axis  of  synnnetry  of  the 
curve. 

AVhon  x=2(i,  // =  0  ;  as  x  decreases,  //  increases;  and  when 
x  =  0.  ?/ =  x  .  If  .!•  is  negative,  or  greater  than  '2  a.  //  is  imagi- 
nary.     The   shape-  of  the  curve   is   something   like  that  in  the 


Chap.  X.] 


AREAS. 


145 


figure,  the  axis  of  T being  an  asymptote.    Tlie  area  Itetween  the 
curve  and  the  asymptote  is  tlien  either 


A  =  ■>  i  'jjdx      Ol-      .1  =  2  i  'j;hi  ; 

by  tlic  first  formula. 

by  tlie  second. 

■'-■   (hi 


Jo  y-  -\-  A  a- 


EXAMPI.KS. 

(1)  Find  the  area  between  the  curve  y-{x-  +  a-)  =  (rx^  and  its 
asymptote  y  =  n.  Ans.   A  =  -la'-. 

(2)  Find  the  area  between  y"('2a— .r)  =  .r'' and  its  asymptote 
x  =  2  a.  Aus.   A  =  :)  -((-'. 

f3)   Fincl  tlie  area  bounded  by  the  curve  y-=— —       '  and 

.       ~  ,    ,  (I  —  X 

Its  asymptote  x  =  a. 


Ans.   A  =  '2crf\  +  -\ 


130.  If  the  coordinates  of  the  points  of  a  cnrxc  iin-  ex- 
pressed in  terms  of  an  auxiliary  variable,  no  new  dillit  iilty  is 
presented. 

Take  the  case  of  the  circle  x'-  -\-  y-  =  a'\  whicli  may  be  writttMi 

.r  =  ((c()s0  i 

y  =  a  sin  cf>  ) 

dy  =  (icofi<f>(J(fi. 


The  whole  area    A  =  a-  I  cos-(f)d4>  =  tto-. 


146  INTEGRAL   CALCULUS.  [Art.  13L 


Examples. 

(1)  The  whole  area  of  an  ellipse  ^  — "'^^^'P  l  jg  ^^5, 

y  =  0  sin  <t> ) 

(2)  The  area  of  an  arch  of  the  cycloid  is  d-n-a^. 

(3)  The  area  of  an  arch  of  the  companion  to  the  cycloid 
=  a$,  y  =  a{\  —  cos^)  is  2 ttcl^. 


131.  If  we  wish  to  find  the  area  between  two  curves,  or  the 
area  bounded  by  a  closed  curve,  the  altitude  of  our  elementar}- 
rectangle  is  the  difference  between  the  two  values  of  y,  which 
correspond  to  a  single  value  of  x.  If  the  area  between  two 
curves  is  required,  we  must  find  the  abscissas  of  their  points  of 
intersection,  and  they  will  be  our  limits  of  integration  ;  if  the 
whole  area  bounded  by  a  closed  curve  is  required,  we  must  find 
the  values  of  x  belonging  to  the  points  of  contact  of  tangents 
parallel  to  the  axis  of  Y. 

Let  us  find  the  whole  area  of  the  curve 

ci^y-  +  h'-x^  =  a-lroi?, 

or  a*y-  =  lrx-{a^—  x^) . 

The  curve  is  sj-mmetrieal  with  reference  to  the  axis  of  X,  and 
passes  through  the  origin.  It  consists  of  two  loops  Avhose  areas 
must  be  found  separately.  Let  us  find  where  the  tangents  are 
parallel  to  the  axis  of  Y. 

b 


(ly  _  h    ((-  —  2  x^ 


(Ix 


'a-  —  X- 


=  tanr. 

:  -  when  tanr  =  »,  that  is,  when  x  —  ±  a. 
2 

yi  =  2 -  fx  V((^  -  x'.dx  +  2  -,  Cx  Va-  -  x'.dx  =  | ab. 


Cn\T.  X.]  AREAS.  147 

Again  ;   find  the  -whole  area  of  (^y  —xy  =  a^  —  7?. 
y  =  x±  ^Id-  —  ar, 
A  =  C{y'-  y")  dx  =  p2  V^?^^' .  dx. 

nits  of  integratic 
^  _  Va-  —  x^  q:  x 


To  find  the  limits  of  integration,  we  must  see  wiiere  t  =  -• 

2 


dx 


y/ar  —  XT 


=  X    when  x  =  ±  a. 


^  =  2  I    V(("  —  .r .  (?a*  =  tto- 

KXAMPLES. 


3r  ^fl  -4'  X^ 

(1)   Find  the  area  of  the  loop  of  the  curve  ?/-=      ^ 


a  —  X 
Ans.    2  a-  (  1  — 


4 

(2)  Find  the    area   between   the    curves   y-  —  4ax=0    and 

^  ^l»s. 

3 

(3)  Find  the  whole  area  of  the  curve  x^  +  v^  =  a}.  Ans.  fjra^ 

4  a- 

(4)  Find  the  area  of  a  loop  of  d-y^  =  x\a-  —  a-) .   ^-l«.s-.  -^— 

o 

(5)  Find  the  whole  area  of  the  curve 

2 /  (a^  +  ar')  -  4  ay  (a^  -  ar)  +  ((«-  -  ar)=  =  0. 

Ans.  (rVA-^Y 
132.    We  have  seen  that  in  polar  coordinates 
A  =  ^\     rd<f>. 

Let  us  try  one  or  two  examples. 

(a)  To  find  the  whole  area  of  a  circle. 

The  polar  ecpuition  is         /•  =  a. 

A  =  \j''''<rd4>  =  7raK 


148  INTEGRAL  CALCULUS.  [Art.  132. 

(/')  To  find  Ihe  aiva  of  the  cardioide  r=  2ff(l  —  cos<^). 
^1  =  ^  I  4  a- ( 1  —  cos  <i>Yd(i>  =  2  o- 1  ( 1  -  2  cos  <^  +  cos' (l(f>)d<ji, 

(c)   To  find  the  area  between  an  arch  of  the  epicycloid  and  the 
circumference  of  the  fixed  circle. 


X  —  {(I  +  h)cosO  —  b  cos  - 

b 

y  =  (a  +  /^)sin  d-b  sin  ^^^  6  { 


1. 


We  can  get  the  area  I)ounded  hy  two  radii  vectores  and  the 
arch  in  question,  and  subtract  the  area  of  the  corresponding 
sector  of  the  fixed  circle. 

Changing  to  polar  coordinates, 

X  =  r  cos  <^, 
?/  =  ?•  sin  <^. 
AVe  want  ^  j  i^d<^. 

y 

tan  (f>  =  ■-' 

^       X 

2  ,  J  .      xfhi—  ydx 
sec'cf>dcf>=      ■' y      ; 

1)nt,  since  x  =  r  cos  <^,     sec  <f>  =  -; 

X 

hence  i^^^xdy-ydx^ 

and  r-  d(j>  =  xdif  —  ydx  ; 

dx  =  (a  +  b)  f-  sin  ^  +sin  ^-^-±-^  ^^  f/^, 

r/v  =  (a  -\-b)f  i:ose  -  cos'-L±J!e]de. 

xdy  -  ydx  =  (a  +  b)  {a  +  2  b)f\  -  cos  -  Aw  =  i'^dcfi. 

Our  limits  of  integration  are  olniouslv  0  and  — ^. 


niAi'.  X.]  AREAS.  149 


Hence 


A  =  ^{((-i-b){((+-2 h )J '  "  / 1  -  fos - 6\  de. 


A  =  ^{a+b){u+2b), 


is  the  area  of  the  sector  of  the  epieycloiil.     Siihtnut  the  :uv;i  of 
the  circular  sector  Trab,  and  we  get 

-1   =   IT 

a 
as  the  area  in  question. 

(d)   To  find  the  area  of  a  loop  of  the  curve  r  =  a-cos  2  </>. 
For  any  value  of  cf)  the  values  of  /•  are  equal  with  opposite 
signs.     Hence  the  origin  is  a  centre. 

When<^  =  0,  r=±(i;    as  0  increases,  r  decreases  in  length 

till  <^  =  -,  when  r  =  0  ;  as  soon  as  0>  -,  r  is  iniaginarv.     If  d> 

4  4 

decreases  from  0,  /•  decreases  in  length  until  0=  — -.  when  r  =  0  ; 

and  when  <^ ,  r  is  imaginary.     To  get  the  area  of  a  looj), 

then,  we  nuist  integrate  from  (f)=  —^  to  cf>  =  -. 

A  =  ^jr^^H  =  ^  crf'^vo^  2  ct>.dct>  =  ^'. 

EXAMPLKS. 

(1)    Find  the  area  of  a  sector  of  the  parabola  r 


1  4-  cos</> 
(2)   Find  the  area  of  a  loop  of  the  curve  rcoscf>  =  irs'm  :5  0. 


Itis.    '-^ '"  log 2. 


I 

(3)  Find  the  whole  area  of  the  curve  r  =  (i(vos'2<f)  +  sin2<^). 

Ans.    7r«*. 

(4)  Find  the  area  of  a  loop  of  the  curve  rcos0  =  m  cos2<^. 

..l„.s-.    (2 -")"». 

(5)  Find  the  area  between  ;-=a(sec</>  +  tan<^)  ami  its  asynip- 
tote  rcos<^  =  2((.  .,,,^     (--hAc'- 

\'^        J 


150 


INTEGRAL   CALCULTS. 


[Akt.  138. 


133.  Wliiu  the  eqiuiUou  of  a  curve  is  given  in  rectangular 
coordinates,  we  can  often  simplify  the  problem  of  finding  its  area 
by  transforming  to  polar  coordinates. 

For  example,  let  us  find  the  area  of 

{x'  +  y-)-  =  ia-x^-\-4bh/. 
Transform  to  polar  coordinates. 

'/•■'  =  4  r(a2cos2<^  +  b-  H\n^<t>), 

t'^  =  i  {(i^ cos- <f) -i- b- Hin- <f)) , 

A=2  i  (d^cos-c^  +  b- sin- (li)(lcf>  =  2  7r(a-  +  b-). 

Examples. 
(1)   Find  the  area  of  a  loop  of  the  curve  {x-  +  y-y  =  4  a-x^ij^. 

Arts.  — — 


(2),  Find  the  whole  area  of  the  curv 


x^      ?/2      1  /.r2      w^' 
a*^b*     c\a'^by 

Ans.  ^(a'  +  b'-). 
2ab 

(3)  P'ind  the  area  of  a  loop  of  the  curve  y^  —  S  axy  -f  x^  =  0. 

2  ' 


A71S, 


134.   Tlie  area  between  a  curve  and  its  evolute  can  easily  be 
found  from  the  intrinsic  equation  of  the  curve. 

It  is  easily  seen  that  the  area 
hounded  by  the  radii  of  curvature 
at  two  points  infinitely  near,  by 
the  curve  and  by  the  evolute,  dif- 
fers from  ^p^f/r  by  an  infinitesimal 
of  liigher  order.  The  area  bounded 
l)y  two  given  radii  vectorcs,  the 
curve  and  the  evolute,  is  then 


CllAP.  X.] 


151 


Hence 


uT'^)- 


For  example,  the  area  between  a  cycloid  anil  its  evoliite  is 


Let 


=  8a^  I  cos-TfZr. 


0     and     T.  = 


=«"! 


2cos-tcZt  =  'Zira^. 


Examples. 

(1)  Find  the  area  between  a  circle  and  its  evohite. 

(2)  Find  the  area  between  the  circle  and  its  involute. 


r^ 


Ilolditch's   TJieorem. 

135.   If  a  line  of  fixed  length  move  with  its  ends  on  any  closed 
curve  which  is  always  concave  toward  it,  the  area  between  the 
^  curve  and  the  locus  of  a  given 

point  of  the  moving  line  is 
equal  to  the  area  of  an  el- 
lipse, of  which  the  segments 
into  which  the  line  is  divided 
by  the  given  point  are  the 
semi-axes. 

Let  the  figure  represent 
the  given  curve,  the  locus 
of /^  and  the  envelope  of  tiie 
moving  line. 

Let  .l/*=a  and  Pn  =  f>, 
and  let  Cli  =  p,  C  Iteing  the 
point  of  contact  of  the  moving  line  with  itis  envelope.  I>ct 
AB  =  a  +  b  =  c. 


152 


INTEGRAL   CALCULUS. 


[Art.  135. 


The  area  between  the  first  euvAe  nnrl  the  .second  is  the  area 
between  the  fir.st  curve  and  the  envelope,  minus  the  area  between 
the  second  curve  and  the  envelope. 

Let  6  be  the  angle  wliich 
the  moving  line  makes  at 
any  instant  with  some  fixed 
direction.  Let  the  figure 
represent  two.  near  positions 
of  the  moving  line  ;  A^,  the 
angle  ])etween  these  posi- 
tions, being  the  principal  in- 
finitesimal. 

PB  =  p,  P'B'  =  p  +  Ap. 

The  area  PBB'P'P  diflfers 
from  ^p~d6  bv  an  infinitesi- 
mal of  higher  order  than  tlie  first. 

^p-cW  is  the  area  of  PBMP,  and  differs  from  PP'NB  by  less 
than  the  rectangle  on  PJ/and  PQ,  which  is  of  higher  order  than 
the  first,  by  I.  Art.  L53.  But  PP'NB  differs  from  PP'B'B  by 
less  than  the  rectangle  on  BX  and  NB\  which  is  of  higher  order 
than  the  first,  since  NB\  which  is  less  than  I*P'-\-\p,  is  infini- 
tesimal and  A6  is  infinitesimal. 

The  area  between  the   first  curve  and  the  envelope   is  then 

hW  ;    or,  since   we  can  take  PP'A'A  just  as  well   for  our 


elementary  area,  ^  I  {c  —  p)^cW. 

U}'dO=^r{c-prdO; 


Hence 


whence 


2  c  rpf/^  =  2: 

^2rr 

I  pdd  =  ir<. 


(1) 


The    area   between    the   second    curve   and   the   envelope    is 

,2Tr 


hj^ip-hyae. 


Chap.  X.]  AREAS.  1, 

The  area  between  the  first  curve  and  the  .second  is  then 


l.y  (1), 


=  bjjj(W-b'7r 
=  Trbc  —  b'-  TT 
=  7r6(o  -hb)  —  b-ir, 
A  =  77Ub.  (2) 

which  is  the  area  of  an  elUpse  uf  wiiich  a  and  b  are  serai-axes. 


Q.  E.  D. 


Examples. 


(1)  If  a  line  of  fixed  length  move  with  its  extremities  on  two 
lines  at  right  angles  with  each  other,  the  area  of  the  locus  of  a 
given  point  of  the  line  is  that  of  an  ellii)se  on  the  segments  o( 
the  line  as  semi-axes. 

(2)  The  result  of  (1)  holds  even  when  the  fixed  lines  are  not 
perpendicular. 

Areas  by  Double  Integration 

136.  If  we  take  x  and  ?/  as  the  coihdinates  of  any  point  P 
within  our  area,  x  and  y  will  be  independent  variables,  and 
we  can  find  the  area  bounded  by  two 
given  cui-ves,  //  =  ./>  and  // =  F.r. 
by  a  double  integration.  Supposo 
the  area  in  question  divided  into 
slices  by  lines  drawn  parallel  to  the 
Mxis  of  y,  anil  these  slices  sulttli- 
\  ided  into  parallelograms  by  lines 
drawn  parallel  to  the  axis  of  X. 
The  area  of  any  one  of  tiie  small 
parallelograms  is  A'/A.r.  If  we 
keep  X  constant,  ami  take  thr  sum 
of  these  rectangles  from  y=fx  to  »/  =  ^'f-  we  sliali  get  u  result 
differing  from  the  area  of  the  corresponding  slice  by  less  than 


154 


TNTEGRAL   CALCULUS. 


[Art.  137. 


2Aa;A?/,  which  is  infinitesimal  of  the  second  order  if  Ax  and  Ay 
are  of  the  first  order. 
Hence 


^x.dy  =  Ax  t  dy 


is  the  area  of  the  sUce  in  question.  If  now  we  take  the  limit  of 
the  sum  of  all  these  slices,  choosing  our  initial  and  final  values 
of  X,  so  that  we  shall  include  the  whole  area,  we  shall  get  the 
area  required. 

Hence  A=  i  '(  i  dy\dx. 

In  writing  a  double  integi'al,  the  parentheses  are  usually  omit- 
ted for  the  sake  of  conciseness,  and  this  formula  is  given  as 

<Fz 

dydx, 

the  order  in  which  the  integrations  are  to  be  performed  being  the 
same  as  if  the  parentheses  were  actualh*  -written. 

If  w'e  begin  by  keeping?/  constant,  and  integrating  with  respect 
to  X,  we  shall  get  the  area  of  a  slice  formed  by  lines  p^allel  to 
the  axis  of  X,  and  we  shall  have  to  take  the  limit  of  the  sum  of 
these  slices  varying  y  in  such  a  way  as  to  include  the  whole  area 
desired.     In  that  case  we  should  use  the  formula 

A  =  I       I  dxdy. 

137.  For  example,  let  us  find  the  area  bounded  l)y  the  para- 
bolas ?/^  =  4  ax  and  ar'  =  4  ay. 

The  parabolas  intersect  at  the  origin  and  at  the  point  (4  a,  4o). 


I      ^^dydx.    or    A^X    j/'^^^t ' 

\a  ta 

C-yiax 


4  ((X ; 

4o 


I       J    dydx  =  j       (  V4  ax  —  —  ) dx  =  —  a^ 


4  a 


The  second  formula  gives  the  same  result. 


Chap.  X.J 


AUEAS. 


Examples. 

(1)  Find  the  area  of  a  rectangle  by  doiil)le  inte^jration  ;  of  a 
[)arallelogram  ;  of  a  triangle. 

(2)  Find  the  area  between  the  parabola  ?/-  =  <tx  and  the  circle 

y-  =  '2  ax  —  a~.  ,         ,  /ts 

Ans.  2f  - 


4  :}  / 


(3)   Find  the  whole  area  of  the  curve  (?/  —  mx  —  c)-  =  a-  —  x^. 

Ans.  irir. 

138.    If  we  use  polar  coordinates  we  can  still  fnid  our  areas 

by  double  integration. 

Let  r  =/({>  and  r  =  F<^ 
be  two  curves.  Divide  the 
area  between  them  into 
slices  ])y  drawing  radii 
vectores ;  then  subdivide 
these  slices  by  drawing 
arcs  of  circles,  witli  tlie 
origin  as  centre. 

Let  P,  with  coordinates 
r  and  <^,  be  any  point 
within    the    space    whose 

area  is  sought.    The  curvilinear  rectangle  at  1*  has  the  base  rA<^ 

and  the  altitude  Ar ;  its  area  ditlers  from  rA<;^Ar  by  an  infniitcsi- 

mnl  of  higher  order  tlian  /-A^A/-. 

Tlie  area  of  anv  slice  as  nha'b'  is  j  r\<f>(h\  0  and  A</)  being 
constant,  that  is  A(^  j  rdr.  The  whole  area,  the  Uiuit  of  the 
sum  of  such  slices  is  .1=1       |  rdr(l<i).  ( 1 


y 

X, 

e^ 

^«' 

1 

" 

slices  is  .1  =  I      j  nlr(l<l>. 

st  sum  our  rectangles, 
le  area  of  pf<''f' 

i    d<f>.    and    A=  i      )  nlifxlr. 


Or  we  may  first  sum  our  rectangles,  kee[>ing  /•   luiehauLred. 
and  we  get  as  the  area  of  efi'\f 

(2) 


156 


INTEGRAL   CALCULUS. 


[Art.  138. 


It  must  be  kept  in  mind  that  r  in  (1)  and  (2)  is  the  radius 
vector  of  any  point  within  the  area  sought,  and  not  of  a  point 
on  the  boundary. 

For  example,  the  area  between  two  concentric  circles,  r  =  a 
and  r  =  h,  is 

.1  =  r  Crdcfydr  =  C      Cnlrd^  =  ir^a^  -  6^) . 

Again,  let  us  find  the  area  between  two 
tangent  circles  and  a  diameter  through  the 
point  of  contact. 

Let  a  and  h  be  the  two  radii, 

?'=2acos<^  (1) 

and  r=26cos^  (2) 

are  the  equations  of  the  two  circles. 

0    J2  h  cos  0  .70  2 

If  we  wish  to  reverse  the  order  of  our  integrations  we  must 
break  our  area  into  two  parts  by  an  arc  described  from  the  origin 
as  a  centre,  and  with  2  6  as  a  radius  ;  then  we  have 


=  f      Crd<i>dr+  C      Crdxfxir 

Jo       J  r  Jib    .'0 

cos-' 26 

r(  COS"'— -  — cOs~'— -  )f7r-|-  j?-cos~^T-dr 
0     V         ^a  2b)         J2b  2a 


|(a^-6^). 


EXAMPLK. 

Find  the  area  between  the  axis  of  X  and  two  coils  of  the 
spiral  /•=  a<f). 


^c^C^o/if      "    )    "Ic/t 


^ 


ClIAP.  XI.] 


AREAS    OF   SUKFAGE&. 


157 


CHAPTER    XL 


AREAS    OF    SURFACES. 


T_^ 


V^'*? 


Surfaces  of  Revolution. 

139.  If  a  plane  curve  y=fx  revolves  about  the  axis  of  X,  the 
area  of  the  surface  generated  is  the  limit  of  the  sum  uf  the  areas 
generated  by  the  chords  of  the  infinitesimal 
arcs  into  which  the  whole  arc  may  be  broken 
up.  Each  of  these  chords  will  generate  the 
surface  of  the  frustum  of  a  cone  of  revolution 
if  it  revolves  completely  around  the  axis ; 
and  the  area  of  the  surface  of  a  frustum 
of  a  cone  of  revolution  is,  by  elementary 
Geometry,  one-half  the  sum  of  the  circum- 
ferences of  the  bases  multiplied  by  the  slant  height.  The  frustum 
generated  by  the  chord  in  the  figure  will  have  an  area  differing 
by  infinitesimals  of  higher  order  from  w  (y  +  y  -\- Ay) ^s  or  from 
2iryds.     The  area  generated  by  any  given  arc  is  then 


S  =  'Itt  i      yds 

through  an  an 
le  surface  gen« 

yds. 


[1] 


If  the  arc  revolves  through  an  angle  $  instead  of  makinf 
complete  revolution,  the  surface  generated  is 


[2] 

It  must  be  noted  that  [1]  and  [2]  will  give  a  positive  value 
for  S  if  the  generating  curve  lies  wholly  above  the  axis  of  X  at 
the  start,  and  a  negative  value  for  S  if  it  lies  wholly  below  the 
axis  of  X  at  the  start.  If  the  curve  happens  to  cross  the  axis 
of  X  between  the  points  whose  ordinates  are  //o  and  y,,  [1]  and 
[2]  give  not  the  area  of  the  surface  generated  by  the  curve  in 
question,  but  the  difference  between  the  areas  generated  by  the 


158 


INTEGRAL   CALCULUS. 


[Art.  140. 


portion  originally  above  the  axis,  and  the  portion  originally 
below  the  axis. 


Example. 
Show  that  if  the  arc  revolves  about  the  axis  of  Y,  S  =  2ir  j  ; 


xds. 


140.    To  find  the  area  of  a  cylinder  of  revolution. 

Take  the  axis  of  the  cylinder  as  the  axis  of  X  Let  a  be  the 
altitude  and  b  the  radius  of  the  base  of  the 
cylinder.      The  equation   of    the    revolving 


line 


y  =  b; 
dy=0, 
ds=  VfZa^  +  di/ 

S^27rf\jdx: 


=  dx ; 

27rO&, 


or  the  product  of  the  altitude  by  the  circumference  of  the  base. 
Again,  let  us  find  the  surface  of  a  zone. 
The  equation  of  the  generating  circle  is 


,       adx 
ds= ; 

y 


r 


adx  =  2  a-TT  (.r,  —  x^) . 


If  X(,  =  —  a  and  Xi  =  a,         S  =  4a^Tr. 

Hence  the  surface  of  a  zone  is  the  altitude  of  the  zone  multi- 
plied by  the  circumference  of  a  great  circle,  and  the  surface  of 
a  sphere  is  equal  to  the  areas  of  four  great  circles. 

Again,  take  the  surface  generated  by  the  revolution  of  a 
cycloid  about  its  base. 


x  =  n6  —  asin^l 
y  =  a  —  a  cos  6  J  ' 
ds  =  add  V2(l  -cos^) , 
r  r"a-V2.(l  -co^efdd  = 


by  Art.  105  ; 


Chap.  XL]  AREAS   OF   SUIIFACES.  l.V.» 


EXAMI'LES. 

(1)  The  area  of  the  surface  generated  l)y  the  revohition  of 
the  ellipse  ^  _l  .V"  _  i 

or      b'- 

about  the  axis  of  X  is  2Tv(ihi'\J\  —  e- -\ 

about  the  axis  of  Y  is  'Itrtr  (\  +  ^  ~  ^'"  log  J— ^Y 

,  o      a-  —  Ir 

where  e-  = -• 

cr 

(2)  t'ind  the  area  of  the  surface  generated  by  the  revolution 
of  the  catenary  about  the  axis  of  X ;  about  the  axis  of  Y. 

(3)  The  whole  surface  generated  by  the  revolution  of  the 
tractrix  about  its  asymptote  is  47ra-. 

(4)  The  area  generated  by  the  revolution  of  a  cycloid  about 
its  vertical  axis  is  87ra-(7r  — |). 

(5)  The  area  generated  by  the  revolution  of  a  cycloid  about 
the  tangent  at  its  vertex  is  ^Tra-. 

(6)  The  area  generated  by  the  revolution  of  the  curve 
jc*  +  y^  =  a*  about  its  axis  is  ^ttci^. 

141.  If  we  know  the  area  generated  by  the  revolution  of  a 
curve  al)out  any  axis,  we  can  get  the  area  generated  by  the 
revolution  about  any  parallel  axis  by  an  easy  transformation  of 
coordinates. 

Given  the  surface  generated  ]»y  the  arc  from  .So  to  .s,  al)out 

s ox,  to  find  the  area  generated  by 

^">N^         the    same    arc    when    it    revolves 

A''  about  O'X'. 

Let  «S'  be  the  surface  about  OX, 
and   6"  about  O'X'. 
■         ^Ve  have 


S  =  27r  f'yds,     »S"=  2 tt  C'i'ds'. 


160  INTEGRAL   CALCULUS.  [Akt.  141 

By  Anal.  Geom.,  x  =  x\ 

y  =  !h  +  y'- 

Hence  (Jx  =  dx\     chj  =  df/',     ds  =  ds\ 

and        S=2-!r  i  {ijo  +  y') d-s  =  2 Tri/o{-^i  —  .s,,)  +  iw  |  y^ds, 

=  2  7r?/o(.s', -So)4-'S". 

There  fore  S '  =  ,S'  -  2  tt.Vo  ( Si  -  -So)  .  [  1  ] 

Si  —  .S)  is  the  lengtli  of  the  revolving  curve  ;  2  tt?/,,  is  the  cir- 
cumference of  a  (;ir(;le  of  which  y^  is  the  radius.  Hence  the  new 
area  is  equal  to  the  old  area  minus  the  area  of  a  cylinder  whose 
length  is  the  length  of  the  given  arc  and  whose  base  is  a  circle 
of  which  the  distance  between  the  two  lines  is  radius. 

In  using  this  principle  careful  attention  must  be  paid  to  the 
sign  of  yo,   and  it  must  be  noted   that   the   original   formula 

s  will  always  give  a  negative  ^'alue  for  the  area  of 


=  2  TT  (  yd 

Js,, 


the  surface  generated,  if  the  revolving  arc  starts  from  below  the 
axis  ;  and  hence,  that  the  surface  generated 
by  the  revolution  of  any  curve  about  an 
axis  of  symmetry  will  come  out  zero. 

As  an  example  of  the  use  of  the  princi- 
ple, let  us  find  the  surface  of  a  ring. 

Let  a  be  the  distance  of  the  centre  of   

the  circle  from  the  axis,  and  h  the  radius  of 

the  circle.     vSince  the  area  generated  b}-  the 

revolution  of  the  circle  about  a  diameter  is  zero,  the  recjuired 

area  is 

'lirhrlira  —  \irah. 

EXAMI'LK. 

Find  the  area  of  the  ring  gciu'rated  by  the  revolution  of  a 
cycloid  about  any  axis  parallel  to  its  l)ase. 

Ans.    o  =  4  uhTT[  it  -\ 

oh 


CiiAi'.  XI.]  AIIKAS    OF    srUKAC^ES. 

142.    If  we  use  polar  for»rcliiKxtes, 

r  sin  <t).ds. 


where  ds  =  vdr''^  +  rd^iK 

For  example  ;  let  us  fiud  the  area  of  the  surface  generated  !)>• 
the  revolution  of  the  upper  half  of  a  eardioide  about  the  hori- 
zontal axis. 

r  =  2a(l  —  cos(^)  ; 

dr  ='2  a  sin<^.rf<^, 

ds2=8a-(l-eos<^)f?<^2^ 

S  =  -2tv  Ca-^2 a-{  1  -  cos <^) ^ sin <f>.d(f>. 

P^XAMPLES. 

(1)  Find  the  surface  of  a  sphere  from  the  polar  equation. 

(2)  Find  the  surface  of  a  paraboloid  of  revolution  from  the 
polar  equation  of  the  parabola 

m 


1  —  cos  <f> 


C;/lin  drica  I  S  u  rfaces. 

143.  If  a  cylindrical  surface  is  generated  bv  a  line  which  is 
always  parallel  to  the  axis  of  Z,  the  area  of  the  portion  bounded 
by  two  positions  of  the  generating  line,  the  plane  of  A'l',  and 
any  curve  whose  projection  on  the  plane  of  XZ  is  given,  is 
easily  found. 

Let  ABCD  be  the  cylindrical  area  required. 


162 


INTEGRAL   CALCULUS. 


[AitT.  143. 


Let  y=fx  (1) 

be  the  equation  of  AB,  the  line  of  iutersection  of  the  surface 
with  the  phvue  XY \   and  let 

z  =  Fx  (2) 

be  the  equation   of  CiT),,  thi;  projection  of  CD  on  the  plane 
of  XZ. 

If  x,y,z  are  tlie  coordinates  of 
any  point  P  of  CZ>,  the  i-equired 
area  is  evidently  the  limit  of 
the  sum  of  rectangles,  of  which 

p  pi  pit  put  Jg       ^^j^y       Qj^Q  -pjjg        j^j.^,g 

of  pp'pop'i'  differs  by  an  in- 
finitesimal of  higher  order  than 
ds  from  zds,  and   therefore   the 

required  area  -S"  =  |    zds. 

x,z  are  the  coordinates  of  Pi,  and 
satisfy  (2),  and  ds=  'sjdsr  +  djf 
where  a*,  //  are  the  coordinates  of 
F  and  satisfy  (1). 


We  have,  then, 


r'z\ldx'-\-dy\ 


[3] 


For  exami)lo,  let  AB  be  the  quadrant  of  a  circle,  and  let  the 
projection  of  the  required  area  on  the  plane  of  XZ  be  the  quad- 
rant of  an  equal  circle,  so  that  the  surface  required  is  one-eighth 
of  the  surface  of  a  groin. 


Here 

x-  +  y'  =  a\ 

(4) 

nd 

(5) 

ds-'^dx^  +  di/'-'-^dx-      ^'^^     , 

and 


-Ja'-; 


Chap.  XL] 


AREAS   OF   SURFACES. 


103 


Therefore 


(Ir 


Again,  let  us  find  tlie  area  of  the  curved  surface  of  tlie 
portion  of  a  cylinder  of  revohition  included  within  a  splifrical 
surface,  whose  centre  lies  on  the  surface  of  tlie  cylinder,  and 
whose  radius  is  equal  to  the  diameter  of  the  cylinder. 

If  the  centre  of  the  sphere  is  taken  as  the  oritrin,  and  a 
diametral  plane  of  the  cylinder  as  the  plane  of  XZ,  tlie  surface 
required  is  four  times  that  indicated  in  the  figure. 

The  equation  of  the  cylinder  is 

x~  —  ax-\-f  =  0,  {C^) 

and  of  the  sphere 

af  +  y-  +  z--a-  =  0.  (7) 

Subtract  (6)  from  (7),  and  we  get 

z-  +  ax-cr=0  (8) 

as  the  equation  of  a  cylindrical  surface 

perpendicnlar   to   the    plane    XZ,   and 

passing  through  all  the  points  of  intersection  of  (6)  and  (7). 

(8)  is,  then,  the  equation  of  the  projection  on  the  plane  of  XZ 

of  the  line  of  intersection  of  the  given  spherical  surface  and 

the  given  cylindrical  surface. 


From  (G), 
From  (8), 
Hence     S 


ds  =  V(/.tr  +  (I  if  =  — dx  = " 

'^1/  ^^/ax-x" 


z  =  Va^  —  ax. 
-  I    Va-  —  ax  • 
aVn  r"dx_  _    2. 


adx 


__  ay  a  r"  v  a  —  x .  dx 
2  Ju    VxVtt  — X 


and  the  whole  area  requued, 


4S  =  4a\ 


164  INTEGRAL  CALCULUS.  [Akt.  144. 


Examples. 

(1)  Find  the  area  cut  from  the  cylindrical  surface  whose 
base  in  the  phiue  XI'  is  a  quadrant  of  the  curve  x^  -\-yi  =  ai  l)y 
the  plane  x  =  z.  Ans.  fcr. 

(2)  Find  the  area  of  that  portion  of  a  cylindrical  surface 
whose  base  in  the  plane  of  XY  is  a  quadrant  of  the  ellipse 

—  _!-•—  =  1,  and  whose  projection  on  the  plane  of  XZ  is  bounded 

r   .1^'  2   0      ,0   o,  2       ^^  i         e      a6(rt-+rt6  +  &') 

by  the  curve  a^z-  =  6-ar(cr  —  ar) .        Ans.  b  =  — ^^ -■ 

^  ^  ^  3(a  +  6) 

(3)  Let  the  base  of  the  cylindrical  surface  be  a  tractrix, 
whose  vertex  lies  at  a  distance  a  to  the  left  of  the  origin,  and 
whose  asymptote  is  the  axis  of  F,  while  its  projection  on  the 
plane  of  XZ  is  bounded  by  the  parabola  z^  =  —2'nix. 

Ans.  S  =  2a\/2ma. 

(4)  Let  the  base  of  the  cylindrical  surface  be  the  upper  half 
of  a  cycloid,  having  its  vertex  at  the  origin  and  its  base  parallel 
to  the  axis  of  F,  and  at  a  distance  2  a  from  the  origin,  while 
its  projection  on  the  plane  of  XZ  is  bounded  by  the  parabola 
z*=2mx.  j_ns.  S  =  4:a^/am. 


Any  Surface. 

144.  Let  x,  ?/,  z  be  the  coordinates  of  any  point  P  of  the  sur- 
face, and  X  -f-  Ax*,  ?/  -f  Ay,  z  +  \z  the  coordinates  of  a  second 
point  Q  infinitely  near  the  first.  Draw  planes  through  P  and  Q 
parallel  to  the  planes  of  XFand  YZ.  These  planes  will  inter- 
cept a  curved  quadrilateral  PQ  on  the  surface  ;  its  projection  p7, 
a  rectangle,  on  the  plane  of  XZ ;  and  a  parallelogram  p'q^  not 
shown  in  the  figure,  on  the  tangent  plane  at  P,  of  which  ;></  is 
the  projection.  PQ  will  differ  from  ;)Vy'  by  an  infinitesimal  of 
higher  order,  and  therefore  our  required  surface  will  be  the  limit 
of  the  sum  of  the  parallelograms  of  which  p'q'  is  any  one. 


Chap.  XL] 


AREAS   OF    SUUFACKS. 


i»; 


If  yS  is  the  angle  the  tangent  plane  at  P  makes  with  XZ, 
p'q' cos /3=pg  ov  p'f/=jvj  sec /3  =AxAzsec/3,  and  o-,  our  sur- 
face required,  is  equal  to 
the  double  integral 

(r=  I   I  sec ftclxdz 

taken  between  limits  so 
chosen  as  to  embrace  the 
whole  surface. 

The  limit  of  the  sum 
of  the  parallelograms,  of 
which  j/fy'  is  a  type,  will 
be  the  required  surface 
if  the  limit  of  the  sum  of 
the  rectangles,  of  which 
pq  is  a  type,  is  the  pro- 
jection of  the  surface  in 
question  on  the  i)lane  of  XZ ;  so  that  the  values  of  x  and  2 

between  which  we  integrate  in  o- =  |   |  sccfSfLcdz  are  pn-cisely 

those  we  should  use  if  we  were  finding  the  area  of  the  projection 

of  or  by  the  double  integration   I    |  dxdz.  (v.  Art.  130.) 

The  ecpiation  of  the  tangent  plane  at  /'  is 
(X  -  x,)D^J+  {y  -  y,)D,jJ+  {z  -  z„)  I),J=  0,    by  I.  Art.  217, 
(a\),yoi2'o)   standing  for  the  coordinates  of  the  point  of  contact, 
and  f(x,y,z)  =  0  being  the  equation  of  the  surface. 

The  direction  cosines  of  the  perpendicular  from  the  origin  U{)ou 


the  plane  are 


DxJ 


cos/3 


cosy  = 


by  Anal.  Cleom.  of  Three  Dimensions. 


166  INTEGKAL   CALCULUS.  [Art.  U4 

Hence,  dropping  the  accents, 


By  considering  tlie  projections  upon  the  other  coordinate  planes 
we  shall  find 


=ff:Mni±mi±miaya.:      m 


In  each  of  the  formulas  the  derivatives  are  partial  derivatives. 
Let  us  find  the  area  of  the  portion  of  the  surface  of  the  sphere 

x^  +  //"  +  2'  =  cr 

intercepted  b^'  the  three  coordinate  planes. 

DJ=2x, 

<^=r  f-f^I/dz;  (1) 

»/0    ^0  X 


Ji)  Jo  y 


hdx ;  (2) 

c.=  f  j;^...^V.  (3) 

For,  in  the  second  one.  which  agrees  l)est  with  the  figure,  we 
must  take  our  limits  so  that  the  limit  of  the  sum  of  the  projec- 
tions may  he  the  (lundraiit  in  which  the  spliere  is  cut  by'the 


Chap.  XI. j 


AREAS    OF    SURFACES. 


167 


p\nno  XZ  :  and  tho  equation  of  this  section  is  obtained  1)\  lettiuL' 
?/  =  0  in  the  equation  of  the  sphere,  and  is 

ar  +  2-  =  a-, 

whence  z  =  Vtr  —  ar. 

If  we  take  asour  Hniits  in  tho  inte<2;ral  |  -dz  zero  and  Va-— .r 

we  shall  get  the  area  whose  projection  is  a  strip  running  from 

the  axis  of  Xto  the  curve  ;  then,  taking  j  (    |  -  dzylx  from  0  to 

«,  we  shall  get  the  area  whose  projection  is  the  .sum  of  all  the.se 
strips,  and  that  is  our  required  surface. 

2/=  Vo^ 


■""'£i 


V- 

dzdx 


Vo^ 


/: 


dz 


sla^  —  x^  —  z- 
if  we  regard  x  as  constant ; 


Va-  —  .^•^ 


i 


dz 


^\la' 


(T  =  a  \     -  dx  =  —  , 
Jo    2  2 

tlie  required  area.     Formulas  (1)  and  (3)  give  the  same  result. 


145.  Suppose  two  cvlindcrs  of  revolution  drawn  tangent  to 
each  other,  and  perpendicular  to  the  plane  of  a  great  circle  of  a 
sphere,  each  having  the  radius  of  the 
great  circle  as  a  diameter  ;  recpiircd  the 
surface  of  the  sphere  not  indudetl  ]»y 
the  cylinders. 

The  surface  required  is  eight  times 
the  sin-face  of  which  the  shaded  portion 
of  the  figure  is  the  projection. 

If  we  take  the  i)lane  of  the  gnat 
eircif  as  the  plane  of  X  )', 


168 


INTEGRAL    CALCULUS. 


s?  —  ax  -\-  y-  =  0 

is  the  equation  of  the  cylinder,  and 

^  +  y''  +  z'  =  a^ 
of  the  sphere. 

We  have  .=  jj ^1^111^11^21  .lyclc. 

From  (2)  DJ=2x, 

^J=  '2y, 

Hence        o-=  (   \-dydx  =  a\    (     ,       '    ' 


[Aht.  145. 

0) 


(2) 


Our  limits  of  integration  for  y  are  Vaa;  —  .x-^  and  Vc(^  —  x^;  for 
X  are  0  and  a. 


sld' 


= sm" 

2  ''a  +  a; 


find 


Let 
and 


I  sin"^A  — ^ — -.f/.r  we  must  integrate  by  parts. 
Jo  ya  +  x 


u  =  sm 
dv  =  rf-r ; 

V  =  X, 

dn  = 


'>{a  +  x)\x 


.dx; 


Vrt  r  V.1 


I  snr\l  .f?x  =  .rsin  'a I  _ 

./  \«  +  A-  \((+.f        2  ./  a 


-\-x 


.dx. 


C'livr.  XI.J  ARKAS    OF    srUFACKS.  1G9 

Let  ii:  =  y/x  ;     2  nxlw  =  dx 

r^Jx.dx  rid-dw  /V  a     \ 

-^  f'  +  -f      V      "^         V«/ 

I  siu~\  — — dx 
Jo  \a  +  x 

Z  4         4  2 

8o-  =  8a^  is  the  whole  surrjice  in  question. 

146.  Let  us  find  the  area  of  the  curved  surface  of  a  right 
cone  whose  base  is  the  curve  x^  +  2/*  =  a^  Jvud  whose  altitude 
is  c. 

If  we  take  the  vertex  of  the  cone  as  the  origin  of  coordinates, 
and  its  axis  as  the  axis  of  Z,  the  equation  of  its  curved  surface 
is 

x^  +  f^^C^)\  (1) 


and  the  projection  of  the  surface  on  the  plane  of  Xi'is  hounded 
by  the  curve 

xi  +  f,  =  ah  (2) 

From  (1)  we  get 

DJ  \         a^      xlyi     ' 

where  x^y  are  the  coordinates  of  any  i)oint  within  tiie  projec- 
tion of  the  base  of  the  cone. 

Since  the  four  faces  of   the  cone  are  equal,   the    required 
surface 

4  r-  /^^«*-^^»  , 


170  INTEGRAL   CALCULUS.  [Art.  UC. 

Let  us  substitute  v^  =  x  aud  tcr'^  =  y,  whence  dx  =  3v^dv 
and  dy  ='6io'dw,  and  we  have 

o-  =  —  I     i  vw  Va^-y  W  -f-  c^ (v^  +  w'f .  diudv  ; 

fi  c/O  »/0 

or,  since  in  a  definite  integral  it  makes  no  difference  what  letters 
we  use  for  the  variables, 

a  =  —  C  Cxy  y/cv'x'y'-\-c\x'  +  ijy .  dydx.  (4) 

The  X  and  y  in  (1),  however,  must  not  bo  confounded  with  the 
X  and  y  in  (3). 

The  integral  in  (4)  is  precisely  that  which  we  should  have  to 
find  if  we  sought  the  area  of  a  surface  of  such  a  nature  that  its 
projection  on  the  plane  of  XY  was  a  quadrant  of  the  circle 
^.2  _|_  ^2  _  ^i^  jj^jj^i  ^|jg  secant  of  the  angle  made  by  the  tangent 
plane  at  any  point  (x,y,z)  of  the  surface  with  the  plane  of  XT 
was  xy \Jo? 3?y'^  +  (?{a?  +  y-f. 

In  the  latter  problem  there  is  nothing  to  prevent  our  re- 
placing X  and  y  in  xy  '\/o?s?y^  +  c^  (x^  +  y'^Y  by  their  values  in 
terms  of  r  and  <^,  the  polar  coordinates  of  any  point  of  the 
projection  a.-^  +  ?/^  =  al,  and  dividing  this  projection  into  polar 
elements  instead  of  rectangular  elements,  and  then  integrating 
between  the  limits  which  we  should  use  if  we  were  finding  the 

area  of  the  projection  by  the  formula  -^1=1    |  rd(f>dr. 
We  have,  then, 

o-  =  —  (      I    r-  sin  <^  cos  ^  Va^?**  sin-  <f>  cos- <^  +  c;-?'* .  rdrdA^ 
or 

jr 

<r=  —  I      I     /'''  sin  <^  cos  <f>  Va^  sin^  </>  cos^  ^  -h  c^ .  dr  d^, 

«c/0    Jo 

i7=z  C,((  i    s'lucf)  cos (f)  Va^  sin''* <f>  cos'^ <f>  -{-c-.  d<t. 


Cii.vr.  XI.J                       AREAS    OF   SURFACES.  171               j 

Substitute    u  =  s'u\'-(f),   aiul  I 
o-  =  oa  I    Va^M  (1  —  u)  +  c- .  (hi, 

a  =  f  poc  +  (a-'  +  4r)  tan   i-l^l  j 

EXAMTLKS.  ' 

(1)  Find  the   area  includetl    by  the   cylimleis   described  in 

Art.  145  by  direct  integration,    /ux^  fi^   ((.7  ' 

(2)  A  square  hole  is  cut  tlirough  a  sphere,  tlie  axis  of  tlie  | 
liole  coinciding  with  a  diameter  of  the  sphere  ;  find  the  area  of 

the  surface  removed.  .^^i„^4,ju.^a 

(3)  A  c^dinder  is  constructed  on  a  single  loop  of  tlie  Qwxsaf  f  fjt, 
r  —  acosncf),  having  its  generating  lines  perpendicular  to  tbe)  |/J^ 
plane  of  this  curve  ;  determine  the  area  of  the  portion  of  the\^  ^ 
surface  of  the  sphere  ar  +  ?/-  +  2^  =  a^  which  the  cvlinder  inter- ^ 7  -»•( 

(4)  Find  the  area  of  the  portion  of  the  surface  of  the  cone 
described  in  Art.  146  included  by  the  cylinder  .1-  + !/'  =  ^"• 


Ans. 


(5)  Find  the  area  of  the  portion  of  the  surface  of  the  sphere 
a?  +  y-  -\-  z-  =  'lay  cut  out  by  one  nappe  of  the  cone 
Ax^  +  Bz'  =  y\  ^j,^^_  47ra^ 

■   V(l+^)(l-f-5) 

(6)  Find  the  area  of  the  portion  of  the  surface  of  the  sphere 

OiT -\- y- -\- z- =  2  ay  lying  within  the  paraboloid  y=Ax^+Bz'. 

.  2  ira 

Ans.   ■ 

\/AB 

(7)  The  centre  of  a  regular  hexagon  moves  along  a  diameter 
of  a  given  circle  (radius  =  «),  the  plane  of  the  hexagon  being 
perpendicular  to  this  diameter,  and  its  magnitude  varying  in 
such  a  manner  that  one  of  its  diagonals  always  coincides  with 
a  chord  of  the  circle  ;  find  the  surface  generated. 

Ans.  a''{-2  7r  +  '3^'S). 


172  INTEGRAL    CALCULUS. 


[AUT.  147. 


CHAPTER     XII. 

VOLUMES. 

Single  Integration. 


^ 


147.  If  sections  of  a  solid  are  made  by  parallel  planes,  and  a 
set  of  cylinders  drawn,  each  having  for  its  base  one  of  the  sec- 
tions, and  for  its  altitude  the  distance  between  two  adjacent 
cutting  planes,  the  limit  of  the  sum  of  the  volumes  of  these 
cylinders,  as  the  distance  between  the  sections  is  indefinitel}- 
decreased,  is  the  volume  of  the  solid. 

We  shall  take  as  established  by  Geometry  the  flict  that  the 
volume  of  a  cylinder  or  prism  is  the  product  of  the  area  of  its 
base  by  its  altitude. 

It  follows  from  what  has  just  been  said,  that  if,  in  a  given 
solid,  all  of  a  set  of  parallel  sections  are  equal,  the  volume  of 
the  solid  is  its  base  by  its  altitude,  no  matter  how  irregular  its 
form. 

Let  us  find  the  volume  of  a  p3Tamid  having  h 
for  the  area  of  its  base,  and  a  for  its  altitude. 

Divide  the  pyramid  b\-  planes  parallel  to  the 
base,  and  let  z  be  the  area  of  a  section  at  the  dis- 
tance X  from  the  ■vortex. 

"We  know  from  Geometr\-  that  -  =  '^  . 


Hence 


-  or 


Let  the  distance  between  two  adjacent  sections  be  dx ;  then 
the  volume  of  the  cylinder  on  z  is 


and  F,  the  required  voluinc  of 


x-rJx, 


[ho 


nmld, 
3" 


Chap.  XII.] 


VOLUMES. 


17:] 


Precisely-  the  same  reasoning  api)lics  to  any  cone,  viiit-h  will 
therefore  have  for  its  vohiine  one-tliinl  the  product  of  its  Ijase 
by  its  altitude. 

Example. 


[/T^ind 


ind  the  vohmie  of  the  frustum  of  a  pyramid  or  of  a  cone. 


148.  If  a  Hne  move  keeping  always  parallel  to  a  given  plane, 
and  touching  a  plane  curve  and  a  straight  line  parallel  to  the 
plane  of  the  curve,  the  surface  generated  is  called  a  conoid. 
Let  us  find  the  volume  of  a  conoid  when  the  director  line  and 
curve  are  perpendicular  to  the  given  plane. 

,        Divide  the  conoid  into  laminae  by 
planes  parallel  to  the  fixed  plane. 

Let  a^y  be  the  (hstance  between 
two  adjacent  sections,  and  let  x  be 
the  length  of  the  line  \\\  which  any 
\  '  section  cuts  the  base  of  the  conoid  ; 
let  o  be  the  altitude  and  b  the  area 
of  the  I)ase  of  the  figure.  Any  one  of  our  elementary  cylinders 
will  have  for  its  volume  ^ax-A?/,  since  the  area  of  its  triangular 

base  is  ^«x,  and  we  have  V=^n\xdy,  the  limits  of  integra- 
tion being  so  taken  as  to  embrace  the  whole  solid,      j  xdy  ha- 

tween  the  limits  in  question  is  the  area  of  the  base  of  the  co- 
noid ;  hence  its  volume, 


Examples. 

[)  Find  the  volume  of  a  conoid  when  the  director  liiic  and 
curve  am4iot  perpendicular  to  the  given  plane. 

h^  A  woodman  fells  a  tree  2  fci-t  in  diameter,  cutting  lialf- 
way  through  from  each  side.  The  lower  face  of  each  cut  i.s 
horizontal,  and  tlie  upper  face  makes  an  angle  of  4o°  witli  the 
horizontal      How  much  wood  does  he  cut  out? 


174  INTEGRAL   CALCULUS.  [Art.  U'J. 

149.    To  find  tne  volume  of  :m  ellipsoid. 

a-      Ir      <•• 

Take  the  cutting  planes  parallel  to  the  plane  of  XY.     A  sec- 
tion at  the  distance  z  from  the  origin  will  have 

•!^  _l_  •!L'  =  1  _  i'  =  <^'  —  ^'  -' 

a-      U'  c-  c- 

a    1—^ -,  Jt    .^ ^ 

for  its  equation,  and  -y&  —z-  and  -  Vc^— ^-  for  its  semi-axes  ; 

hence  its  area  will  be  ''^—-{r  —  z^). 
r 

Any  of  the  elementary   cylinders   will  have   for  its  volume 
^^(c^  — 2!-)A2;,  and  we  shall  have  for  the  whole  solid 

v=^-^ry-z'^)az. 

V=  ^Trabc. 
If  a,  &,  and  c  are  equal,  the  ellipsoid  is  a  sphere,  and 


U^'^ 


sheet 


Examples. 
Find  the  volume  included  l)etween  an  hyperboloid  of  one 


ih: + ■!l  _  £."  =  1 

((-      Ir       c- 


and  its  asymptotic  cone 

a-      fj-      c- 
Ans.   It  is  equal  to  a  cylinder  of  the  same  altitude  as  the 
solid  in  question,  and  having  for  a  base  the  section  made  by  the 
plane  of^Xr. 

^"(2)   Kind  the  whole  volume  of  the  solid  bounded  by  the  surface 
<  +  f;  +  ^  =  '-  Ans.   ^. 


CiiAi'.  XII. ]^  VOLUMES.  175 

U^  Find  the  volume  cut  from  the  surface 

c       0 

by  a  phme  parallel  to  the  plane  of  (  YZ)  at  a  distance  ((  fioin  it. 

Ans.    Tra-^{bc). 

I  (4)  The  centre  of  a  regular  hexagon  moves  along  a  diameter 
of  a  given  circle  (radius  =  0),  the  plane  of  the  hexagon  being 
perpendicular  to  this  diameter,  and  its  magnitude  varying  in 
such  a  manner  that  one  of  its  diagonals  always  coincides  with 
a  chord  of  the  circle  ;    find  the  volume  generated. 

Ans.    2V3.o\ 
(5)  A  circle  (radius  =  «)  moves  with  its  centre  on  the  cir- 
cumference of  an   equal  circle,  and  keeps  parallel  to  a  given 
plane  which  is  i)erpcndicular  to  the  plane  of  the  given  circle  ; 
find  the  volume  of  the  solid  it  will  generate.  .^   3 

Ans.    fiL(3  7r  +  .S). 
o 

Solids  of  Revolution.     Single  Integration. 

150.  If  a  solid  is  generated  by  the  revolution  of  a  plane  curve 
y  =  f.v  about  the  axis  of  x,  sections  made  by  planes  perpendicu- 
lar to  the  axis  are  circles.  The  area  of  any  such  circle  is  7r»/-, 
the  volume  of  the  elementary  cylinder  is  ttt/'-A-t,  and 

V=  IT  i  irdx 


is  the  volume  of  the  solid  generated. 

For  example  ;  let  us  find  the  volume  of  the  solid  generated  by 
the  revolution  of  one  branch  of  the  tractrix  al)out  the  axis  of  X. 
Here  we  must  integrate  from  a;  =  0  to  x  =  x . 


»/0 


dx. 


AVe  have  dx  =  -  — —  dy         ( A  rt .  1 U2  [2]. ) 

in  tlie  case  of  the  tractrix  : 


176  INTEGRAL   CALCULUS.  [Aur.  15i, 

iR'iice  F=  —  TT  I  >i{<r  —  if) i (iy. 

When  X  =  0,  ^  =  a.   ami  when  x  =  x,  y  =  0. 


V^-^jj]l{ir-f)^ihj  = 


Therefore  F=  -  tt  |  //(rr  -  r) if///  =  ^ 


Examples. 
|>f)   If  the  plane  curve  revolves  aboyt  the  axis  of  F, 

ii^  Th^ volume  of  a  si)]iere  is  \iTi^. 

l^fThc  volume  of  the  solid  formed  b}'  the  revolution  of  a 
cycloid  a)*out  its  base  is  i)Tr-(i^. 

f40  The  curve  y'{'2a  —  x)  =  x^  revolves  about  its  asymptote  ; 
show  tlmt-tlie  volume  generated  is  2Tr-a^. 

(S)  The  curve  xa  +y^  =  os  revolves  about  the  axis  of  X ; 
show  that  the  volume  generated  is  ^^'^^Trct^. 

Solids  of  Revolution.     Double  Integration. 

151.  If  we  suppose  the  area  of  the  revolving  curve  broken  up 
into  infinitesimal  rectangles  as  in  Art.  137,  the  clement  AajAy 
at  an}'  point  P,  whose  coordinates  are  x  and  ?/,  will  generate 
a  ring  the  volume  of  which  will  differ  from  ^-n-y^x^y  by  an 
amount  which  will  be  an  infinitesimal  of  higher  order  than  the 
second  if  we  regard  Ax  and  A?/  as  of  the  first  order.  For 
the  ring  in  question  is  obviously  greater  than  a  prism  having 
the  same  cross-section  A.rA_?/,  and  having  an  altitude  equal  to  the 
inner  circumference  2  Try  of  the  ring,  and  is  less  than  a  prism 
having  A.x-A.?/  for  its  base  and  2ir{y  -|- A?/),  the  outer  circumfer- 
ence of  the  ring,  for  its  altitude  ;  but  these  two  prisms  dilfer  by 
27rAa;(A_7/)'-,  which  is  of  the  third  order. 


Chap.  XII.]  VOLUMES.  177 

'liriidy,  where  the  ii^jper  Hmit  of  iiite^r:itioii   is  the  onli- 

uate  of  the  point  of  the  curve  iinmediately  ahove  /',  and  nui.st  be 
expressed  in  terms  of  .«•  by  the  aid  of  the  equation  of  the  revolv- 
ing curve,  will  give  us  the  elementary  cylinder  used  in  Art.  150. 

The  whole  volume  re(|uired  will  be  the  limit  of  the  sum  of 
these  cylinders  ;  that  is, 

V=-2irr  Cychjdx.  [1] 

If  the  figure  revolved  is  bounded  by  two  curves,  the  required 
volume  can  be  found  by  the  formula  just  obtained,  if  the  limits 
of  integration  are  suitaljly  chosen. 

Let  us  consider  the  following  example  : 

A  paral)oloid  of  revolution  has  its  axis  coincident  with  the 
diameter  of  a  sphere,  and  its  vertex  in  the  surface  of  the  sphere  ; 
recjuired  the  volume  Ijctween  the  two  surfaces. 

Let  y-  =  2mx  (1) 

be  the  parabola,  and      ur  +  y-  —  2 ax  =  0  (2) 

be  the  circle,  which  form  the  paraboloid  and  the  sphere  by  their 
revolution.  The  abscissas  of  their  points  of  intersection  are  U 
and  2 {a  —  in). 

We  have  V=2Tr  j  \  ydydx, 

and.  in  performing  our  fir.st  integration,  our  limits  must  be  the 
values  of  y  obtained  from  equations  (1)  and  (2). 

We  get  r  =  TT  j  [2 (a  —  vi)x  —  ar](?.r, 

and  here  our  limits  of  integration  are  0  and  2(«  —  m). 

Hence  F=  |7r(a  —  m)^  =  — r^ 

(j 

if  h  is  the  altitude  of  the  solid  in  (piestion. 

EXAMIM-KS. 

^l)  A  cone  of  revolution  and  a  paralioloid  <>f  revoluliou  have 
the  same  vertex  and  the  same  base  ;  recjuired  the  volume  be- 
tween them.  ,^^^.     iTU^^  ^^.,^^,.^.  ,^  .^  ^,,^.  ,^^ii,„,,.  „niH.  cone. 


INTEfJKAL   CALCCTLUS.  [Art.  152. 

\2)  Find  the  volume  included  between  a  right  cone,  whose 
vertical  angle  is  30°,  laid  a  sphere  of  given  radius  touching  it 
along  a  circle.  .   „    ttt' 

6 

Solids  of  Revolution.     Polar  Formula. 

l.')2.  If  we  use  polar  coordinates,  and  suppose  the  revolving 
area  broken  up,  as  in  Art.  138;  into  elements  of  which  rd<{>ilr 
is  the  one  at  any  point  P  whose  coordinates  are  r  and  <^,  the 
element  rdcfidr  will  generate  a  ring  whose  volume  will  differ 
from  2  ir7~  sin  (f>d(fidr  b}'  an  infinitesimal  of  higher  order  than  the 
second,  if  we  regard  d(f)  and  dr  as  of  the  first  order ;  for  it  will 
be  less  than  a  prism  having  for  its  base  rd(f)dr,  and  for  its  alti- 
tude 2  TT (r-fdr)  sin (<^ +  d0),  and  gi-eater  than  a  prism  having 
the  same  base  and  the  altitude  2  irr  sin  cf> ;  and  these  prisms 
differ  b}'  an  amount  which  is  infinitesimal  of  higher  order  than 
the  second. 

"We  shall  have  then 

T'=  2  TT  r  fv-^  sin  cf>drd<l>,  [1] 

the  limits  being  so  taken  as  to  bring  in  the  whole  of  the  gener- 
ating area. 

For  example  ;  let  us  find  the  volume  generated  by  the  r.^volu- 
tion  of  a  cardioide  about  its  axis. 

r  =  2  a  ( 1  —  cos  <^) 
is  the  equation  of  the  cardioide  ; 


2  7r  I    I  r»h\(f)drd<^. 


Our  first  integral  must  be  taken  between  the  limits  ;•  =  0  and 
=  2  a  ( 1  —  cos  <f)) ,  and  is 

^{\-GOsct>fs\u<f>d(t>. 
o 

F=— a^Trj  (I  — cos0)^sin<id^, 


,CUAP.  XII.] 


VOLUMES. 


Example. 

right  cone  has  its  vertex  on  tlie  surface  of  a  si)hpre,  and  its 
axis  coincident  with  the  diameter  of  the  si)here  passing  through 
that  point ;  find  the  vohune  connnon  to  the  cone  and  the  spliere. 


Volume  of  ((»>/  Solid.      Tn'jib'   InU'ijrdfion. 

153.    If  we  suppose  our  soU<l  divi(U'(l  into  paranoU)|)ipeds  l>v 
planes  parallel  to  the  three  oo/irdinate  planes,  the  elementary 


parallelopiped  at  any  point  (.r,.?/,2;)  within  the  solid  will  have  for 
its  volume  A.vA/yAri;,  or,  if  we  regard  or,  ?/,  and  z  as  inilependent, 
dxdydz ;  and  the  whole  volume 

V=ff^d,'dyaz,  [1] 

tlie  limits  being  so  chosen  as  to  embrace  the  whole  solid. 

The  integrations  are  independent,  and  may  be  performed  iu 
any  order  if  the  limits  are  suitably  chosen. 

As  it  is  imjiortant  to  have  a  perfectly  clear  conception  of  the 
geometrical  interpretation  of  each  step  in  the  process  of  linding 


180  INTEGRAL  CALCULUS.  [Art.  153. 

a  volume  by  a  triple  integration,  we  will  consider  one  ease  in 

detail. 

Let  the  integrations  be  performed  in  the  order  indicated  by 

the  formuh.  r  r  C 

y=\    1   )  dzdydx. 

If  the  limits  are  correctly  chosen,  our  first  integration  gives 
us  the  volume  of  a  pi'ism  one  of  whose  lateral  edges  passes 
through  any  chosen  point  P,(.'c,^,z)  within  the  solid,  is  parallel 
to  the  axis  of  Z,  and  reaches  directly  across  the  solid  from 
surface  to  surface,  while  the  base  of  the  prism  is  the  rectangle 
dydx ;  our  second  integration  gives  the  volume  of  a  right  cylin- 
der whose  base  is  a  plane  section  of  the  solid,  passes  through 
the  i)oint  P,  and  is  parallel  to  the  plane  FZ,  and  whose  altitude 
is  dx ;  and  our  third  integration  gives  the  volume  of  the  whole 
solid. 

The  limits  in  our  first  integration  are,  then,  the  values  of  z 
belonging  to  the  point  in  the  lower  bounding  surface  and  the 
point  in  the  upper  bounding  surface  which  have  the  coordinates 
X  and  y  ;  the  limits  in  the  second  integration  are  the  values  of  y 
belonging  to  the  two  points  in  the  perimeter  of  the  projection 
of  the  solid  in  the  plane  of  XF  which  have  the  coordinate  x\ 
and  the  limits  in  the  third  integration  are  the  least  value  and 
the  greatest  value  of  x  belonging  to  points  on  the  perimeter  of 
the  projection  of  the  solid  on  the  plane  of  XK 

It  is  easily  seen  from  what  has  just  been  said  that  the  limits 
in  the  second  and  third  integrations  are  precisely  those  we 
should  use  if  we  were  finding  the  area  of  the  projection  of  the 
solid  by  the  formula  ^  ^ 

A=  \    I  dydx. 

Of  course,  it  is  necessary  to  have  a  clear  idea  of  the  form  of 
the  solid  whose  volume  is  required. 

For  example ,  let  us  find  the  volume  of  the  portion  of  the 
ellipsoid  ^       2       2 

cut  off  l)y  tlie  coordinale  planes. 


Chai>.  Xll.]  VOLUMES.  181 


and  our  limits  are,  for  2,  0  and  c\|l  — ^— ^;  Ibr  >/.  0  and 

I ?  \      ""       ^' 

6-^1  —  -^;    and  for  x,   0  and  a.      For,  starting  at  anv  point 

{.r.jf.z)  and  integrating  on  tlie  liypotliesis  that  z  alone  varies,  we 
get  a  column  of  our  elementarv  parallelopipeds  having  (ivdi/  as  a 
base  and  passing  through  the  point  {x,y,z).  To  make  this  col- 
umn reach  from  the  plane  XY  to  the  surface,  z  must  increase 
from  the  value  zero  to  the  value  belonging  to  the  point  on  the 
surface  of  the  ellipsoid  which  has  the  coordinates  x  and  // ;  that 

is,  to  the  value  c-^l  —  '—^—jo-     Then,  Integrating  on  the  h}- 

pothesis  that  y  alone  A'aries,  we  shall  sum  these  columns  and 
shall  get  a  slice  of  the  solid  passing  through  {xj/,z)  and  having 
the  thickness  dx.  To  make  this  slice  reach  completely  across 
the  solid,  we  must  let  y  increase  from  the  value  zero  to  the 
greatest  value  it  can  have  in  the  slice  in  question  ;  that  is,  to  the 
value  which  is  the  ordinate  of  that  point  of  the  section  of  the 
ellipsoid  b}-  the  plane  XF  which  has  the  abscissa  x.  The  section 
in  question  has  the  equation 


a'  ^  b' 


therefore  the  required  value  of  ?/  is  ft  \|1  —  ^■ 

\         a- 

Last,  in  integrating  on  the  hypothesis  that  x  alone  varies,  we 
must  choose  our  limits  so  as  to  include  all  the  slices  just  <le- 
scribed,  and  must  increase  x  from  zero  to  a. 


f' 


"— W'-s-;;- 


between  the  limits      0     and 


182 


INTEGRAL   CALCULUS. 


[Art.  153. 


'/V'-m:-"^ 


— E^Yi — — '^ 

-    4   (,        „') 


W-5 


between  the  limits       0     and     b^l 


the  vohime  required 


Trbc  r"f,      x-\,       TTCibc  i/  ,;  > 


Examples. 


UO   Kind  tlie  vohune  obtained  in  tlie  present  article,  perlorni-      1- O' 
mg  the  integrations  in  the  order  indicated  by  tlie  Ibrnuila,  > 

V^C  C  Cdxdiidz. 


^i 


Find  the  volnnie  ent  off  from  the  surface 


z-      V- 
C        h 

by  a  planj^parallel  to  that  of  FZ,  at  a  distance  a  from  it. 

Ans.    ird^-^ibc). 


Find  the  vohnne  enclosed  by  the  snrfiices, 
x'  -\-  y-  =  cz,     or  +  y-  =  ax,     2  =  0. 

(4)   Obtain  the  volume  bounded  l)y  the  surface 
z  =  (i  —  V.f-  +  y- 
and  the  i)lanes  x  =  z     and     x  =  0 


Ans. 


32  c 


2  a'' 


Chap.  XII.]  VOLUMES.  jj^3 

U^*>fY'\Vi^  the  voliinie  of  the  conoid  ])ouii(Ie(l  by  the  surface 

z^j^^  =  (?  and  the  planes  .r  =  0  aiul  x  =  a.  Ans.  ^. 

af  2 

154.    If  we  use  polar  coordinates  we  c^an  take  as  our  element 
of  volume 

r^  sin  ^r//Y/^(/^, 

an  expression  easily  obtained  from  the  element  '2irrs'm<f)drd4i 
used  in  Art.  152. 


Then  V=  f  f  p '  sin  cf^drdcfxie. 


where  the  order  of  the  integrations  is  usually  immaterial  if  the 
limits  are  properly  chosen. 

Examples. 
M^iif)  Find  the  volume  of  a  sphere  by  polar  coordinates. 
tZjl^ind  the  whole  volume  of  the  solid  bounded  by 
(x^  +  ?/-  +  z-f  =27  a^xi/z. 
Suggestion:  Transform  to  polar  coordinates.     Ans.   -(^, 


184 


<=^^c^  /f  ~  -^^^^   -2- 

LNTEGKxVL   CALCULUS. 


[Art.  155 


CHAPTER    XIII. 


CENTKKS    OF     GKAVITY. 

155  The  moment  of  a  force  about  an  axis  perpendicular  to  its 
hue  of  direction  is  the  product  of  the  niacrnitude  of  the  force  bv 
tlie  ,,erpcMKlicular  distance  of  its  line  of  direction  from  the  axis^ 
and  measures  the  tendency  of  the  force  to  produce  rotation 
about  the  axis. 

The  force  exerted  by  gravity  on  any  material  body  is  propor- 
tional to  the  mass  of  the  body,  and  may  be  measured  by  the 
mass  of  the  body.  "^ 

The  Centre  of  Gravity  of  a  bod.y  is  a  point  so  situated  that  the 
force  of  gravity  produces  no  tendency  in  the  body  to  rotate  about 
any  axis  passing  through  tliis  point. 

The  subject  of  centres  of  gravity  belongs  to  Mechanics,  and 
we  shal  accept  the  definitions  and  principles  just  stated  as  data 
for  matiiemat.cal  work,  without  investigating  the  mechanical 
grounds  on  wliich  they  rest.  "itcnanicai 

156.  Suppose  the  points  of  a  ]>ody  referred  to  a  set  of  three 
roctang.1  ar  axes  fixed  in  the  body,  and  let  x^y^z  be  the  coordi- 
nates  of  the  centre  of  gi-avity.  Place 
the  body  with  the  axes  of  X  and  Z 
horizontal,  and  consider  the  tendency 
of  the  particles  of  the  body  to  i)roduce 
rotation  about  an  axis  through  {x,y,z) 
paraUcl  to  OZ,  under  the  influence  of 
gravity.  Represent  the  mass  of  an 
elementary  parallelopiped  at  anv  point 
{x,y,z)  b3-  dm.  The  force  exerted  by 
gravity  on  dm  is  measured  bv  dm,  and 

i™t!!r.,';V"'r"°"  ■'  'f  "'■  "■  ""■  "■•""'  <"■ '"» «•«■«  ™"»n- 

tiatcd  at  1 ,  the  moraeut  o^  tUc  force  exeitol  on  dm  about  the 


CllAl'.  XIII.]  CENTRES    OF    (JUAVITY.  185 

axis  through  C  would  be  {x  —  jc)dm,  anil  this  monicut  would 
rei)resent  the  tendency  of  dm  to  rotate  about  the  axis  in  (jues- 
tion  ;  the  tendency  of  the  whole  body  to  rotate  about  this  axis 
would  be  1{x  —  x)dm.  If  now  we  decrease  dm  indefinitely,  the 
error  committed  in  assuming  that  the  mass  of  dm  is  concentrated 
at  P  decreases  indefinitely,  and  we  shall  have  as  the  true  expres- 
sion for  the  tendency  of  the  whole  body  to  rotate  about  the  axis 
through  C,   I  {x  —  x)dm  ;  but  this  must  be  zero. 

Hence  I  (x  —  x)din  =  0, 

I  xdni  —  ^i  dm  =  0, 

xdm 


f^ 


J 


dm 


[1] 


If  we  place  the  body  so  that  the  axes  of  Y  and  X  are  liori- 
zojital,  the  same  reasoning  will  give  us 


jyd7 


I  din 

and  in  like  manner  we  can  get 


I  zdvi 
^=^4 [3] 


I  dm 

Since  (dm  is  the  mass  of  tiie  whole  body,  if  we  rei)resent  it 

by  Mwe  shall  have  ^ 

I  xdm 


y  =  ' 


M 
M 


I  zdm 


186  INTEGRAL   CALCULUS.  [Art.  157. 

/  Example. 

Show  that  the  effect  of  gravity  in  m.aking  a  body  tend  to  rotate 
about  any  given  axis  is  precisely  the  same  as  if  the  mass  of  the 
body  were  concentrated  at  its  centre  of  gravity. 

157.  The  mass  of  any  homogeneous  body  is  the  product  of 
its  vohune  l)y  its  density.  If  the  body  is  not  homogeneous,  the 
density  at  an}-  point  will  be  a  function  of  tlie  position  of  that 
point.  Let  us  represent  it  by  k.  Then  we  may  regard  (bn  as 
equal  to  kcIv  if  dv  is  the  element  of  volume,  and  we  shall  have 


/ 


xkcIu 

[1] 


I  Kdv 


and  corresponding  formulas  for  f/  and  z. 

If  the  body  considered  is  homogeneous,  k  is  constant,  and  we 
shall  have 

I  xdv       I  xdv 


I  ?/^J^     I  ydv 

I  zdv       I  zdv 


■'="j^='— '  f^^ 


*       V  f^3 


In  any  particular  problem  we   have  only  to  express  dv  in 
terms  of  the  coordinates. 

PUmo  Area. 

158.    If  we  use  rectangular  coordinates,  and  are  dealing  with 
a  plane  area,  where  tiie  weight  is  uniformly  distributed,  we  have 

dr  =  dA=dxdy.  (Art.  1.36). 


Chap.  XIII.]  CENTIIES   OF    (IKAVITY. 

Hence,  by  i:)7,  [2]  and  [r,], 


Oxdxdi/ 
l/  _       ffydxdy 

jjaxay 


If  we  nse  polar  coordinates, 

dv  =  fL-1  =  rd<jidr, 


and 


/J 


r^  cos  <^  d<fidr 


y  = 


I    I  ?-  sin  (^  d0dr 


187 


[1] 


[2] 


For  example  ;  let  ns  find  tlie  centre  of  f/ravitj/  of  the  area  l»e- 
tween  the  cissoid  and  its  asymptote.  From  the  equation  of  the 
cissoid 


r  = 


we  SCO  that  the  curve  is  syinnictricMl  with  respect  to  the  axis 
of  X,  passes  through  the  origin,  ami  has  the  line  x  =  <i  as  an 
asymptote.  From  the  s^-rametry  of  tht-  ana  in  question,  y  =  0. 
and  we  need  only  find  x. 


(i      I  xdifdx        I  xtfdx 
I      I   dydx         I  ydx 


188  INTEGRAL   CALCULUS.  [Art.  158. 


-      J^(a-.r)h  J,{a-x 


dx 
^, ;    by  Art.  G4  [4]. 


■  dx  I — — -  dx 

)(((  —  .<;) 5  Jo  (a— a;)' 

x  =  la. 

As  an  example  of  the  use  of  the  polar  formulas  [2] ,  let  us  fiud 
the  centre  of  gravity  of  the  eardioide 

r=2a{\  —  cos<^). 

Here,  from  the  fact  that  the  axis  of  X  is  an  axis  of  s3-mmetr3', 
we  know  that  ?/  =  0. 

t  I  1^  cof!  </)(?^      — —  I  ( 1  .  _  eos  4> )  ^  cos  0r?<^ 
i  r?-2 d<^  2  a^  n  1  -  cos  (^)-  fZ<^ 

j  (cos </)  —  3  cos^ ^  4-  3  cos'' ^  —  cos^ (f))d(fi  =  —  Jj^ tt  ; 

and        1(1  —  2 cos <^  +  cos-^  )d(f)  =  S tt. 
Hence  x  =  —^a. 


/ 


Examples. 


Show  that  formulas  [1]  hold   even   when  we  use  oblique 
coordinates. 


coormn 

\A.  1 


Find  the  centre  of  tiravitv  of  a  sciiincut  of  n  paral)ola  cut 
off  by  any  chord. 

Ans.    .r=fa.     i/^O.      If  tiie  axes  are  the  tangent  parallel 
to  the  chord  and  the  diameter  bisecting  the  chord. 


CENTRES   OF   GRAVITY. 


189 


Find  the  centre  of  gravity  of  llie  area  hounded  l)y  the  si-nii- 
cubical  uarabolti  ay-  =  x"  and  a  double  ortlhiate.      .l/(.s.  x  =  ix. 


cuDicalparaoola  ay  =  x'  and  a 
ly^  Find  the  centre  of  gravit 


ravity  of  a  senii-eUipse,  the  bisecting 
line  being  any  diameter.- 

Ans.  If  the  bisecting  diameter  is  taken  as  the  axis  of  Y,  and 

the  conjtlgate  diameter  as  the  axis  of  X,    x  =—,     ]/  =  0. 

37r 

i>.   Find  the  centre  of  gravity  of  the  curve  y-  =  h-^!~^- 


Ans.  x  =  \a. 
Iml  the  centre  of  gravity  of  the  cycloid.       .    y^ 

xhis.   x  =  aTr,    y  =  |a. 

//^  Find  the  centre  of  gravity-  of  the  lemniscate  r  =  (rcos2<^. 


\y^" 


A  -  7rV2 

Ans.   x  = a. 

H 


Find  the  centre  of  gravity  of  a  circular  sector. 
Ans.   If  we  take  the  radius  bisecting  the  sector  as  the  axis 

a  sin  a 


of  X,  and  rg^^resent  the  angle  of  the  sector  by  2a,  x  =  f 

ind  the  centre  of  gravity  of  the  seirment  of  an  ellipse  cut 

h 


drantal  chord.       Ans.   a;  =  f 


//  =  S 


the  centre  of  gravity  of  a  quadrant  of  the  area  of  the 


y-h  =  al. 


Ans.  X 


v  =  m.- 


159.    If  we  are  dealing  with  a  homogeneous  solid  formed  by 
the  revolution  of  a  plane  curve  about  the  axis  of  X,  we  have 


(Iv  =  'Iirydydx. 


(Art.  i:.i  [1]; 


Hence,  by  Art.  157  [2], 


j   jxijdxdy 
ffydxdy 


[1] 


-."V' 


\ 


OH 


,0 

\ 


190  INTECUAL   CALCULUS.  [AuT.  159. 

If  we  use  i)ol:ir  coordinates, 

dv  =  '2Tn-sin<pdr<Icl>.  (Art.  ir,2  [1].) 

I    I  ?'''sin(/)Cos<^(?rf7</» 

Hence  x  =  '^  ^^ [2] 

I    i  r- sin  (jidn I (fi 

For  example  :  let  us  lind  the  centre  of  gravity  of  a  hemisphere. 
The  eciuution  of  the  revolvinji,'  curve  is  x- +  if- —  ir.  „ 


ion  of  the  revolving  c 
I      I  r''sin</)  cos  (fid<jid 


If  we  use  i^olar  coordinates  the  equation  of  the  revolving  curve 
IS  /•  =  a. 


Here         ..  —  -  ,    „ 


J       I  rs'incfiddx 
0     Jit 


/  Examples. 

Vi .  Find  the  centre  of  gravity  of  the  solid  formed  by  the  revolu- 
tion of  the  sector  of  a  circle  about  one^f  its  extreme  radii. 

/      Avs.   a;  =  |a  cos-^/8,  where  /3  is  the  angle  of  the  sector. 

V^.  Find,  the  centre  of  gravity  of  the  segment  of  a  paraboloid 
of  revolut(tf)n  cut  oti' by  a  plane  perpendicular  to  the  axis. 

/  Ahs.   x  =  |(f,  where  x  =  a  is  the  plane. 

lii.  Find  the  centre  of  gravity  of  the  solid  formed  by  scooping 
out  a  cone  from  a  given  i)araboloid  of  revolution,  the  bases  of 
the  two  volumes  being  coincident  as  well  as  their  vertices. 

Aus.   The  centre  of  i^ravitv  bisects  the  axis. 


r 


CuAi'.  XIII. j  CENTKES   OF   CRAVITY,  191 

/4.   A  cardioide  is  made  to  revolve  about  its  axis  ;    lind  llie 
cent^of  gravitv  of  the  solid  generated.  .l/(.s.  .?  =  —  ?«. 

^o.   Obtain  formulas  for  the  centre  of  gravity  of  any  lionio- 
geneous;g<51id. 

)6/^ind  the  centre  of  gravity  of  the  solid  bounded   l>y  the 

sHrface  z-  =  xy  and  the  five  planes  a;=0,,2/=0,  z=0^  x=a,  i/=h. 

Ans.   a-='"|«,  y=fb,   2  =  ^'j«i/y4. 

160.    If  we  are  dealing  with  the   arc  of  a   plane  curve,  tlie 
formulas  of  Art.  167  reduce  to 

I  ads 
x  =  '^,  [1] 

■r 

,     '  I  yds 


fi. 


Examples. 

l/i.  Find  the  centre  of  gravity  of  an  arc  of  a  circle,  taking  the 
diameter  bisecting  the  arc  as  the  axis  of  X  and  the  centre  as  tiie 
origin. 

iJir  Find  the  centre  of  gravity  of  tlie  arc  of  the  curve  x'i-\-y^=n\ 
between  two  successive  cusps.  Ans.  x  =  y=  j((. 

3.  Find  the  centre  of  gravity  of  the  arc  of  a  semi-cycloid. 

.1//.S.   .r  =  (7r-^)*(,     »/==-$ a. 

4.  Find  the  centre  of  gravity  of  the  arc  of  a  catenary  cut  off 
by  any  horizontal  chord. 

Ans.x  =  0,     y  =  -— — -^,     where  2.S  is  the  lentrth  of  the  arc. 
2  .s- 

/>f  Obtain  formulas  for  the  centre  of  gravity  of  a  surface  of 
revolution,  the  weight  being  uniformly  distributed  over  the 
surftice. 


INTEGRAL   CALCULUS.  [Art.  161 


6.   Find  the  centre  ol'  gravity  of  uny  zone  of  a  sphere. 
Ans.   The  centre  of  gra\ity  bisects  the  hne  joining  the  centres 
of  the  basgf  of  the  zone. 

A  cardioide  revolves  about  its   axis  ;   find  the  centre  of 
gravity  of  tjje  surface  generated.  Ans.  x  =  —  Ya**  ('- 

ind  the  centre  of  gravity  of  the  surface  of  a  lieniisphero 
when  the  density  at  each  point  of  the  surface  varies  as  its  per- 
pendicular distance  from  the  base  of  the  hemisphere. 

^^  Ans.  x  =  ^((. 


^.  Find  the  centre  of  gravity  of  a  quadrant  of  a  circle,  the 
density  at  any  point  of  which  varies  as  the  vfh  jiower  of  its 
distance  from  the  centre.  ^,jg    jj;  _  -^  _  ^'  +'2  2a 

■    '  ■  7t  +  3"  tt' 

\^.  Find  the  centre  of  gravity  of  a  hemisphere,  the  density 
of  which  varies  as  the  distance  from  the  centre  of  the  sphere. 

Ans.  x  =  |a. 

Properfies  of  C'lildin. 

161.  I.  If  a  plane  area  revolve  about  an  axis  external  to 
itself  through  any  assigned  angle,  the  volume  of  the  solid  gene- 
rated will  be  equal  to  a  prism  whose  base  is  the  revolving  area 
and  whose  altitude  is  the  length  of  the  path  described  by  tlie 
centre  of  gravity  of  the  area. 

II.  If  the  arc  of  a  plane  curve  revolve  about  an  external  axis 
in  its  own  plane  through  any  assigned  angle,  tlie  area  of  th<! 
surface  generated  will  be  equal  to  that  of  a  rectangle,  one  side 
of  which  IS  the  length  of  the  revolving  curve,  and  the  other  tiie 
length  of  the  path  descril)ed  by  its  centre  of  gravity. 

First ;  let  the  area  in  question  revolve  about  the  axis  of  X 
through  an  angle  0.  The  ordinate  of  the  centre  of  gravit}-  of 
the  area  ni  (piestion  is 


,r.A 


]/(l.cdi/ 
y  =  ''^-'i '  l>.y  A rt .  1 58  [  1  ] . 


C  Cdxibi 


CiiAl'.  XIII.]  CKNTUKS    OF    CItAVlTY.  ll»3 

The  loniitli  of  the  path  th-scrihcd  hy  the  centre  of  gruvity 
(")  I    I  ydxdy 

The  voUiine  gem-vated  is 

y=    ^-^  I   i  >/<^->^di/,  by  Alt.  151. 

Hence  V=T/@i   i  dxdij. 

But  I    I  dxd)/  is  the  revolving  area,  and  the  first  tlieorem  is 
established. 

"We  leave  the  proof  of  the  second  theorem  to  the  student. 

EXAMPttS. 

*}-rTind  the  surface  and  volume  of  a  sphere,  regarding  it  as 
generated  by  the  revolution  of  a  semicircle,  a  ^       -^^ 

y^.   Find  the  surface  and  volume  of  the  solid  generated  by  the 
revolution  of  a  cycloid  about  its  base. 

l^i\¥'\nd  the  volume  and  the  surface  of  the  ring  generati'd  by 
the  revolution  of  a  circle  about  an  external  axis, 

Ann.     V=2iT-a-h.     S^Airab,     where  h  is  the  distance  of 
the  centre  of  the  circle  from  the  axis. 

lAf^nd  the  volume  of  the  ring  generated  by  tiie  revolution  of 
an  ellipse  about  an  external  axis. 

Ans.  V=27r\dj(:,  where  c  is  the  distance  of  the  centre  of  the 
ellipse  from  the  axis. 


/h^  -1 


/v 


r"  ^  "f^ 


\  ^^-r 


194  IJS'TEGUAL   CALCULUS.  [Art.  WL 


LINE,    SURFACE,   AND   SPACE   INTEGRALS. 

162.  Any  variable  which  depends  for  its  value  solely  upon 
the  position  of  a  point,  as,  for  example,  any  function  of  the 
rectangular  or  polar  coordinates  of  the  point,  may  be  called 
a  point- function. 

A  point-function  is  said  to  be  continuous  along  a  given  line 
if  its  value  changes  continuously  as  the  point,  on  whose  position 
the  function  depends  for  its  value,  moves  along  the  line  ;  it  is 
said  to  be  continuous  over  a  given  surface  if  its  value  changes 
continuously  as  the  point  is  made  to  move  at  pleasure  over  the 
surface ;  and  it  is  said  to  be  continuous  throughout  a  given 
space  if  its  value  changes  continuously  as  the  point  is  made  to 
move  about  at  pleasure  within  the  space. 

163.  If  a  given  line  is  divided  in  any  way  into  infinitesimal 
elements,  and  the  length  of  each  element  is  multiplied  by  the 
value  a  given  point-function,  which  is  continuous  along  the  line, 
has  at  some  point  within  the  element,  the  limit  approached  by 
the  sum  of  these  products  as  each  element  is  indefinitely  de- 
creased, is  called  the  line  integral  of  the  given  function  along 
the  line  in  question. 

If  a  given  surface  is  divided  in  any  way  into  infinitesimal 
elements  such  that  the  distance  between  the  two  most  widely 
separated  points  within  each  element  is  infinitesimal,  and  the 
area  of  each  element  is  multiplied  by  the  value  a  given  point- 
function,  which  is  continuous  over  tlie  surface,  has  at  some 
point  within  the  element,  the  limit  approached  by  the  sum  of 
tliese  products  as  each  element  is  indefinitely  decreased,  is 
called  the  surface  integral  of  the  given  function  over  the  surface 
in  question. 


CiiAi'.  XIV.]  L.IXE,    SURFACE,    STAGE    INTKCiUALS.  1%') 

If  a  given  space  is  divided  in  :iuy  way  into  infinitesimal 
elements  such  that  the  distance  between  the  two  most  widely 
separated  points  within  each  element  is  infinitesimal,  and  ti»e 
volume  of  each  element  is  multiplied  by  the  value  a  given  point- 
function,  which  is  continuous  throughout  the  space,  has  at 
some  point  within  the  element,  the  limit  approached  by  the 
sum  of  these  products  as  each  element  is  indefinitely  decreased, 
is  called  the  sjxice  integnd  of  the  given  function  throughout 
the  space  in  question. 

It  is  easily  seen  that  the  line  integral  of  unity  along  a  given 
line  is  the  length  of  the  line  ;  that  the  surface  integral  of  unity 
over  a  given  surface  is  the  area  of  the  surface  ;  and  that  the 
space  integral  of  unity  throughout  a  given  space  is  the  volume 
of  the  space. 

In  the  chapter  on  Centres  of  Gravity  we  have  had  numerous 
simple  examples  of  line,  surface,  and  space  integrals. 

1G4.  That  the  value  of  a  line,  surface,  or  space  integral  is 
independent  of  the  position  in  each  element  of  the  point  at 
which  the  value  of  the  given  function  is  taken  can  be  proved 
as  follows  :  The  distance  apart  of  any  two  points  in  the  same 
infinitesimal  element  is  infinitesimal  (Art.  163),  therefore  the 
values  of  a  continuous  function  taken  at  any  two  points  in 
the  same  element  will  differ  in  general  by  an  infinitesimal ;  the 
products  obtained  by  multiplying  these  two  values  by  the  mag- 
nitude of  the  element  will,  then,  differ  by  an  infinitesimal  of 
higher  order  than  that  of  the  element ;  therefore,  in  forming 
the  integral  either  of  these  products  may  l)e  used  in  place  of 
the  other  witliout  changing  the  result.      (I.  Art.  KJl.) 

165.  The  line  integral  of  a  function  along  a  given  line  is 
absolutely  independent  of  the  maimer  in  whicii  the  line  is 
broken  up  into  infinitesimal  elements,  and  is  equal  to  the  K-ngtli 
of  the  line  multiplied  by  the  mean  value  of  the  function  along 
the  line;  the  vieaii  value  of  the  function  being  defined  as  fol- 
lows: Suppose  a  set  of  points  uniformly  distriluited  along  the 


196  1^TKGKAL   CALCULUS.  [Akt.  165. 

line,  that  is,  so  (listiil)iited  that  the  mimbor  of  points  iu  any 
portion  of  the  line  is  proportional  to  tlie  length  of  the  portion  ; 
take  the  value  of  the  function  at  each  of  these  points  ;  divide 
the  sum  of  these  values  by  the  number  of  tlie  points  ;  and  the 
limit  approached  by  this  quotient  as  the  number  of  the  points 
is  indefinitely  increased  is  the  7nean  value  of  the  given  function 
along  tlie  line  ;  and  this  mean  value  is  in  general  finite  and 
determinate. 

To  prove  our  proi)ositiou,  we  have  only  to  consider  iu  detail 
the  method  of  finding  the  mean  value  in  question.  Let  the 
number  of  points  in  a  unit  of  lengtli  of  the  line  be  k.  Then, 
no  matter  how  the  line  is  broken  up  into  infinitesimal  elements, 
the  numl)er  of  points  in  each  element  is  A;  times  the  lengtii  of  the 
element.  Since  any  two  values  of  the  function  corresponding  to 
points  iu  the  same  element  differ  by  an  infinitesimal,  in  finding 
our  limit  we  may  replace  all  values  corresi)onding  to  points  in 
the  same  element  by  any  one  ;  hence  the  sum  of  the  vaUies  cor- 
responding to  points  in  the  same  element  may  be  replaced  by  one 
value  multiplied  by  the  number  of  points  taken  iu  that  element,* 
that  is,  this  sum  may  be  replaced  by  k  times  the  product  of  one 
value  by  the  length  of  the  element ;  and  the  sum  of  the  values 
corresponding  to  all  the  points  taken  in  the  line  may  be  replaced 
by  k  times  the  sum  of  the  terms  obtained  by  multiplying  the 
length  of  each  element  by  the  value  of  the  function  at  some 
point  within  the  element.  When  we  divide  this  sum  by  tlie  whole 
number  of  points  considered,  that  is,  by  k  times  the  length  of 
the  line,  the  A;'s  cancel  out,  and  the  required  mean  value  reduces 
to  the  limit  of  the  numerator  divided  by  the  length  of  the  line, 
and  the  limit  of  tiie  numerator  is  the  line  integral  of  the  func- 
tion along  the  line.  Therefore  the  line  integral  is  the  mean 
value  of  the  function  multiplied  by  the  length  of  the  line. 

The  same  proof  may  l»e  given  for  a  surface  integral  or  for  a 
space  integral.  The  former  is  the  product  of  the  area  of  the 
surface  by  the  mean  value  of  the  function  over  the  surface ; 
the  latter  is  tiie  volume  of  the  space  multiplied  by  the  mean 
value  of  the  function  throughout  the  space ;  and  both  are  inde- 


Chap.  XIV.]  LINE,    SUKFACE,    SPACE    INTECIIALS.  lit? 

pendent  of  the  way  in  winch  the  surface  or  si)ace  may  he  divided 
into  intiuitesimal  elements. 

166.  If  the  line  along  which  the  integral  is  taken  is  a  plane 
cui-ve,  it  is  easy  to  get  a  geometrical  representation  of  the 
integral.  For,  if  at  every  point  of  the  line  a  perpendicular  to 
the  plane  of  the  line  is  erected  whose  length  is  equal  to  the 
value  of  the  function  at  the  point,  the  line  integral  n-quired 
clearly  represents  the  area  of  the  cylindrical  surface  containing 
the  perpendiculars  if  the  values  are  all  of  the  same  sign,  and 
represents  the  difference  of  the  areas  of  the  portions  of  the 
cylindrical  surface  which  lie  on  opposite  sides  of  the  line  if  the 
values  of  the  function  are  not  all  of  the  same  sign. 

A  similar  construction  shows  that  a  surface  integral  over  a 
plane  surface  may  be  represented  by  a  volume  or  by  the  differ- 
ences of  volumes.  Consequently,  in  each  case  if  the  function 
is  finite  and  continuous,  the  integral  is  finite  and  determinate. 

167.  As  examples  of  line,  surface,  and  space  integrals,  we 
will  calculate  a  few  moments  of  inertia. 

The  moment  of  inertia  of  a  body  about  a  given  axis  may  be 
defined  as  the  space  integral  of  the  i)rodiK"t  of  tlie  density  at 
any  point  of  the  body  by  the  square  of  the  distance  of  the  point 
from  the  axis;  the  integral  being  taken  throughout  the  space 
occupied  by  the  body. 

If  the  body  considered  is  a  material  surface  or  a  material 
line,  the  integral  reduces  to  a  surface  integral  or  to  a  line 
integral. 

In  the  examples  taken  below  the  Ixxly  is  supposed  to  be 
homogeneous. 

(«)  The  moment  of  inertia  of  a  circumference  alK)ut  a  given 
diameter. 

Using  polar  coordinates  and  taking  the  diameter  as  our  axis, 

/  =  I     a-  sin"  (/)  •  k(i(i^  =  ka^n 
=  iMa\  [1] 


198'  INTEGRAL   CALCULUS.  [Art.  167. 

if  I  is  the  moment  of  inertia,  and  a  the  radius,  k  the  density, 
and  M  the  mass  of  the  circumference  in  question. 

{h)  The  moment  of  inertia  of  the  perimeter  of  a  square  about 
an  axis  passing  through  the  centre  of  the  square  and  parallel 
to  a  side. 

7=2  Cy-My  +  2  Cc^Mx 

=  |i^/«^  [2] 

if  '2  a  is  the  length  of  a  side. 

(c)  The  moment  of  inertia  of  a  circle  about  a  diameter. 

Xa     rt2TT 
I     r  sin-  (f> .  knl({>dr  =  ^  k-n-a* 

=  iMa\  [3] 

(d)  The  moment  of  inertia  of  a  square  about  an  axis  through 
the  centre  of  the  square  and  parallel  to  a  side. 

/=  I      I    y^kdxdy=^ka* 

=  iMaK  [4] 

(p)  The  moment  of  inertia  of  the  surface  of  a  sphere  about 
a  diameter. 

I     a^  sin" ^ .  A;a-  sin  (jidcftdO  =  ^  kira* 
./o 

=  ^MaK  [5] 

(/)  The  moment  of  inertia  of  the  surface  of  a  cube  about  an 
axis  parallel  to  an  edge  and  passing  through  the  centre. 

7=1  f"   C\a-  +  z'')kdxdz  -J-  2  f"   C\y^  +  z^)kdydz 

=  "j'  ka'  +  \£  ka* 

=  ^iMa\  [6] 


Chap.  XIV.]        LINE,   SrUFACE,   SPACE   INTEGRALS.  IW 

{g)  The  moment  of  inertia  of  ii  si)liero  alxnit  a  diameter. 

1=  i      i     (  ■»"  siu-  ^ .  kr  siu  <t>  drd<j>(ie  =  y\  ^-Tra^ 

=  iMcr.  [7] 

(h)  The  moment  of  inertia  of  a  cube  about  an  axis  through 
the  ceutre  and  parallel  to  au  edge. 


/  =  £"  f"  C\y^  +  z^)MxcJych  =  -'/A-'a* 


=  |3/a^  [8] 

Examples. 

Find  tlie  moments  of  inertia  of  the  following  bodies : 


)    Of   a  straight  line  about  a   jierpondicular   through   an 
extremity  ;  about  a  perpendicular  through  its  middle  point. 

y^  Ans.  iMl';  ^\Ml\ 

i^z)  Of  the  circumference  of  a  circle  about  au  axis  through 
its  centpe^perpendiciilar  to  its  plane.  A)is.  Mo?. 

JJ(P)  Of  a  circle  about  an  axis  through  its  centre  perpendicular 
to  its  nlttne.  .  Ans.  ^Ma?. 


L<4)  Of 


througli  its 


^A)  Of  a  rectangle  whose  sides  are  2  a,  26,  about  an  axis 
through  its  centre  perpendicular  to  its  plane  ;  about  au  axis 
through  its  centre  parallel  to  the  side  2h. 

Ans.  i3/(«-  +  6-);  \Ma\ 

Of  an  ellipse  about  its  major  axis  ;  about  its  minor  axis  ; 
about  an^axis  through  the  centre  peipendicular  to  the  plane  of 
theejKl^e.  Ans.  \MU'\  \Mcr\  \  M {(.r -{- U') . 

^^6)  Of  an  ellipsoid  about  the  axis  a.  Ans.    I  .V(//- -f- r). 

]J(f)   0f  a  rectangular  parallelopiped  about  an  axis  through 
the  c^trc  parallel  to  the  edge  2«.  Ans.   \M{U'  -\-r). 

\/8)   Of  a  segment  of  a  ]Kira])ola  about  tlie  principal  axis. 

Ans.  i-V6-,  where  2  6  is  the  breadtii  of  the  segment. 


H- 


200 


INTEGRAL   CALCULUS. 


[Akt.  H>8. 


168.  If  u,  D^u,  and  DyU  are  finite,  continuous,  and  single- 
valued  for  all  points  in  a  given  ^^Zarte  surface  hounded  by  a 
closed  curve  T,  the  surface  integral  ofDy^xx  taken  over  the  surface 
is  equal  to  the  line  integral  of  ncosa  taken  around  the  li'hole 
bounding  curve,  where  a  is  the  angle  made  with  the  axis  of  X 
by  the  external  normal  at  any  point  of  the  boundary. 

This  may  be  Ibniiulated  thus  : 

C  CD^udxd;/  =  Cu 


cos  a.  ds. 


[1] 


Let  the  axes  be  chosen  so  that  the  surface  in  question  lies  in 
the  first  quadrant,  and  divide  the  projection  of  T  on  the  axis 
of  1''  into  infinitesimal  elements  of  which  any  one  is  dy. 


On  each  of  these  elements  as  a  base  erect  a  rectangle  ;  and 
since  T  is  a  closed  ciirve,  each  of  these  rectangles  will  cut  it 
an  even  number  of  times. 

Let  us  call  the  values  of  u  at  the  points  where  the  lower  side 
of  any  one  of  these  rectangles  cuts  T,  «,,  u.,,  »3,  u^,  etc.,  re- 
spectively ;  tlie  angles  wliich  this  side  makes  with  the  exterior 
normals  at  these  points,  u,,  a^,,  u;,,  a^,  etc.  ;  and  the  elements 
which  the  rectangle  cuts  from  T,  rf.s,,  (Zsji  ds^,  da^,  etc. 

It  is  evident  that  whenever  a  line  parallel  to  the  axis  of  X 
cuts  into  the  surface  bounded  by  T,  the  corresponding  value  of 
a  is  obtuse  and  its  cosine  negative  ;   that  whenever  it  cuts  out, 


CliAV.   XIV.]         LINE,    SURFACE,    SI'ACE    INTEGKALS.  201 

a  is  acute  and  its  cosine  positive  ;  aud  tliat  any  value  of  a  18 
the  angle  which  the  contour  T  itself  makes  at  the  point  in  ques- 
tion with  the  axis  of  Y  if  we  suppose  the  contour  traced  by  a 
point  moving  so  as  to  keep  the  bounded  surface  always  on  the 
left  hand. 

We  have  then  approximately, 

dll=  —dSi  •COSui  =  f/.s'o-  C()Sa._,=  —ds.^-  COSuj  =  f?S4-  C0Sa4='".      [2] 

If,  now,  in  |    |  D^udxdy   we   perform    the    integration   with 
respect  to  .r,  and  introduce  the  proper  limits,  we  shall  have 

j    \D, ndxd;/  =  Cd;/ {—  n^  +  u.,—  //,.j  +  »4 •  •  • )  ;  [3] 

and  the  second  member  indicates  that  we  are  to  form  a  quantity 
corresponding  to  that  in  parenthesis  for  every  rectangle  which 
cuts  T,  to  multiply  it  by  the  base  of  the  rectangle,  and  then  to 
take  the  limit  of  the  sura  of  the  results  as  all  the  bases  are 
nidefinitely  decreased. 

By  [2], 

f??/ (—?/ 1    -I-    ?^,  —    ?<..;+».,•••  ) 

=  Vi  cos  ai  dSi  4-  H.,  cos  a^  r/.s\,  +  ii..  COS  ttg  dS:^  +  "4  <^'t>«  "1  ''•"''i  +  •  •  •  ;    [-^^l 

and  the  limit  of  the  sum  of  the  values  any  one  of  which  is 
represented  by  the  second  member  of  [4]  is  clearly  |  aconads 
taken  around  the  whole  of  T. 


F^XAMI'I-K. 

iPp^Tve  that  under  the  conditions  stutiMJ  in  ilic  last  article 

C  Cj)^Ndxd>/=  Cncosfi.ds, 

where  /3  is  the  angle  made  with  the  axis  of   1'  liy  tlie  exti-rior 
normal. 


202  INTEGRAL  CALCULUS.  [Akt.  169. 

169.    As  an  illustration  of  the  last  proposition,  let  us  find  the 
centre  of  gravity  of  a  semicircle. 

We  have  ^  ^  lu  ffy^^^^-  ( ^ ) 

But  we  may  write  y  =  D^{xi/).     Hence,  by  Art.  1G8, 

y  =  -^ffy^^-''^^!^  =  17  J -^'^  cos  ads 

=  —  {  i   acos^asin^cos</)ad^  4-  j    x.O.cos-'dx] 
3f\Jo  J -a  2        ) 

_     Ti        23p._4« 

2 

which  agrees  with  the  result  of  Ex.  8,  Art.  158. 

As  a  second  example,  we  shall  find  the  moment  of  inertia  of 
a  circle  about  a  diameter. 
We  have 

I=k\    I  y-dxdy  =A:  |  xy"^  cos  4> .  ds 

=  ki    a  cos  (}>  a- shr<j>(;oscf>  ad  (f> 

=  Aa^  f  sin>  cos-<f>dcf>  =  -  Tra^  =  -Ma% 
Jo  4-1 

which  agrees  with  the  result  of  (*•),  Art.  167, 

EXAMI'LKS. 

^^)   Find  the  centre  of  gravity  of   a  semicircle,  using  the 
theorem  |    i  D^udxdy  =  i  ucosfS .ds. 

Y/(2)   Find  the  moment  of  inertia  of  a  circle  about  an  axis 
through  its  centre  perpendicular  to  its  plane,  using  the  principles 

I   I  D^udxdy  =  J  v  cos  a .  ds    and     |    |  DytaLxdy  =  |  ?<  cos/? .  ds. 


Chap.  XIV.]         LINE,   SURFACE,   SPACE    INTEGRALS.  208 

170.  Since,  as  we  have  seen  in  Art.  168,  a  is  the  angle  which 
the  curve  T  makes  with  the  axis  of  Y;  if  we  trace  the  curve 
so  as  to  keep  the  bounded  space  on  our  left,  it  follows  that 
cosa.cZs  =  dy. 

Hence  |   |  Djidxdy  =  I  udy  ;  [1] 

and  in  like  manner, 

JfD,ndxd>j  =  -f>alx;  [2] 

the  first  integral  in  [1]  and  [2]  being  taken  over  the  bounded 
surface,  aiid  the  second  around  the  bounding  curve. 

For  example,  the  moment  of  inertia  of  a  square  about  au 
axis  through  the  centre  and  parallel  to  a  side  is 

I=kC  Cfdxdij.  (((?)  Art.  1G7.) 

•^  '-  -• '  \    {  y-'f-i-'^U  =  j  xyhhj, 

and  the  last  integral  is  to  be  taken  around  the  perimeter. 
Hence 


l/o 


Example. 


York  Ex.  8,  Art.  1G7,  by  the  aid  of  (2). 

171.  If  U,  D„Tj,  DyU,  and  1)^U  are  finite,  continuous^ 
single-valued  functions  throughout  the  space  bounded  by  a  given 
closed  surface  T,  the  space  integral  of  D,U  taken  throughout  the 
space  in  question  is  equal  to  the  surface  integral,  taken  over  the 
bounding  surface,  of  U  cosa,  where  a  is  the  angle  made  with 
the  axis  of  X  by  the  exterior  normal  at  any  point  of  the  surface. 

This  may  be  formulated  thus : 

r  r  f />.  Udxdydz  =  Cl  Vos a  .  dS.  [  1  ] 


204 


INTEGRAL  CALCULUS. 


[Art.  17L 


The  proof  is  almost  identical  with  that  given  in  Art.  168, 
except  that  for  elementary  rectangle  we  use  elementary  prism. 
We  shall  merely  indicate  the  steps. 


dydz  =  —  rf*S'i  costti  =  dS.,  cosog  =  —  dS._,  cosag  =  ... 
r  r  f/A Udxdydz  =  Cfdydz  [- b\ +  U,-U,--] 
=  the  limit  of  the  sum  of  terms  of  the  form 

Ui  costti .  d<Si  +  U.,  costta .  dS-i  +  U^  cosua .  ^^.S'^  -\ 

=  I  C/cosa.d-S- 

Example. 
Prove  that  under  the  conditions  of  the  last  article 
C  i  CDyUdxdtjdz=  i  Ucos/i.dS, 

and  f  f  Cd,  Udxdydz  =  Cu cos  y .  dS, 

wiierc  P  and  y  are  the  angles  made  witli  the  axes  of  Y  and  Z 
respectively  by  the  exterior  normal  to  the  bounding  surface. 


CllAP.  XIV. 1        LINE,    SURFACE,    SPACE     INTEtlUALS.  205 

172.    As  ail  illustiution,  let  us  liiul  the  centre  of  gravity  of  a 
hemisphere. 
We  have 

X  =  ^  j    I   I  xdxdydz  =  —  |  —  cosa  .fZ-S 

=  - —  I       I    a-  cos-  ^  cos  0  a-  s i n  <^  dd)d6 
2MJo    Jo 

n 

23fJii    Jo         ^       '^   ^ 

a*k       IT     3 

=  --a: 


which  agrees  with  the  result  of  Art.  159. 


Example. 


Find  the  moment  of  inertia  of  a  sphere  about  a  diameter ;  of 
a  cube  about  an  axis  through  the  centre  parallel  to  an  edge. 
IMake  your  work  depend  upon  finding  the  value  of  a  surface 
integral 


20(3  INTEGRAL   CALCULUS.  [Art.  173 


CHAPTER    XV. 

MEAN    VALUE   AND    rilOBABILTTY. 

173.  The  application  of  the  Integral  Calculus  to  questions 
in  Mean  Value  and  Probability  is  a  matter  of  decided  interest; 
Jjut  lack  of  space  will  prevent  our  doing  more  than  solving 
a  few  problems  in  illustration  of  some  of  the  simplest  of  the 
methods  and  devices  ordinarily  employed.  A  full  and  admirable 
treatment  of  the  subject  is  given  in  "Williamson's  Integral 
Calculus"  (London:  Longmans,  Green,  &  Co.);  and  numer- 
ous interesting  problems  are  published  with  their  solutions 
in  "The  Mathematical  Visitor"  and  "The  Annals  of  Mathe- 
matics." 

174.  The  mean  of  n  quantities  is  their  sum  divided  by  their 
number.  If  the  number  of  quantities  considered  is  supposed 
to  increase  indefinitely  according  to  some  given  law,  the  prob- 
lem of  finding  the  limiting  value  approached  by  their  mean 
usually  calls  for  tlie  Integral  Calculus.  The  mean  value  of  a 
continuous  function  of  one,  two,  or  three  independent  variables 
has  been  carefully  defined  in  Art.  165,  and  has  been  proved  to 
depend  upon  a  line,  surface,  or  space  integral. 

(a)  Let  us  find  the  mean  distance  of  all  the  points  on  the 
circumference  of  a  circle  from  a  given  point  on  the  circumfer- 
ence. 

If  we  take  the  given  |)oint  as  origin,  the  distances  whose 
mean  is  required  are  the  radii  vectores  of  points  uniformly  dis- 
tributed along  the  circumference  of  tiie  circle. 

The    recpiirod    mean    is,    therefore,    by    Art.    IGo,    equal    to 


Chap.  XV.]  MEAN    VALUE    AND    IMM  >r.AIULITY.  2U7 

the  quotient  obtained  by  dividing  the  line  integral  of  r  takoii 
around  the  circumference  by  the  length  of  the  cu-ciiinfiTL-nce ; 
that  is, 


I  rds 


'lira 

The  polar  equation  of  the  circle  is 
r='2a  cos  <f> ; 
ds=  2a(Ict>, 

M=  -L  (  '4  a-  cos  <t>  <}ct>  =  — , 
'2iraJ  T  TT 

the  required  mean  value. 

{h)  Let  us  find  the  mean  distance  of  points  on  the  surface 
of  a  circle  from  a  fixed  point  on  the  circumference. 

Here,  by  Art.  I60,  the  required  mean  is  the  surface  integral 
of  r  taken  over  the  circle,  divided  by  the  area  of  the  circle ; 
that  is, 

2..  cos*  ^,^^ 


(c)  The  problem  of  finding  the  mean  distance  of  points  on 
the  surface  of  a  square  from  a  corner  of  the  S(iuare  can  be  sim- 
plified slightly  by  considering  merely  one  of  the  halves  into 
which  the  square  is  divided  by  a  diagonal. 

Here 

^     Z    ..»«'■<:'<• 

M=  -„  I         r-  rdrd4> 

a-J»  J" 


=  |(V-i-Mo.t„niL'). 


208  INTECniAL   CALCULUS.  [Art.  17', 

((/)  As  im  exumple  of  a  (U'vit-o  often  oiiiployotl,  we  shall  now 
solve  the  problem,  To  lliid  the  mean  distance  between  two  points 
within  a  given  circle. 

If  M  be  the  required  mean,  the  sum  of  the  whole  number  of 
cases  can  be  represented  by  (7rr)^iT/,  r  being  the  radius  of  the 
circle ;  since  for  each  position  of  the  first  point  the  number  of 
positions  of  the  second  point  is  proportional  to  the  area  of  the 
circle,  and  may  be  measured  b}'  that  area ;  and  as  the  number 
of  possible  positions  of  the  first  point  may  also  be  measured 
by  the  area  of  the  circle,  the  whole  number  of  cases  to  be  con- 
sidered is  represented  b}'  the  square  of  the  area ;  and  the  sum 
of  all  the  distances  to  be  considered  must  be  the  producit  of  the 
mean  distance  by  the  number. 

Let  us  see  what  change  will  be  produced  in  this  sum  by  in- 
creasing r  hy  the  infinitesimal  dr  ;  that  is,  let  us  find  d{Trr*M). 

If  the  first  point  is  anywhere  on  the  annulus  2  Trr.dr,  which  we 
have  just  added,  its  mean  distance  from  the  other  points  of  the 

circle  is  — ,  by  (b). 
9ir 
Therefore,  the  sum  of  the   new  distances   to  be  considered, 

if  the  first  point  is  on  the  annulus,  is  '-:^.7r7^.2irrdr;  but  the 

second  point  may  be  on  the  annulus,  instead  of  the  first ;  so  that 
to  get  the  sum  of  all  the  new  cases  brought  in  by  increasing 
r  by  dr,  we  must  doul)le  the  value  just  obtained. 

Hence  d{Tr"-r'M)  =  ip  _,.v,,.^ 

TT-d'M  =  J-p  TT  Cr\h  =  Vb^-Tra*, 

45  TT 


17.">.  In  solving  questions  in  Prohahilitii.  we  shall  assume 
that  the  student  is  familiar  with  the  elements  of  the  theory  as 
given  in  "  Todhunter's  Algeljra." 

(a)  A  man  starts  from  the  bank  of  a  straight  river,  and 
walks  till   MooM  ill  a  laiidoiii  dirrction  ;   he  then  turns  and  walks 


CllAP.  XV.]  MEAN    VALUE    AND    I'Kl  UtAKI  MTV.  2Ult 

in  anotluM-  random  diivctioii  ;  wl>:iL  is  tlu'  proliahility  tliat  In-  will 
reach  the  river  by  night? 

Let  ^  be  the  angle  his  Ih-st  course  makes  with  the  river.  If 
the  angle  through  which  lie  turns  at  noon  is  less  than  -  —  -Id. 
he  will  reach  the  river  by  night.  For  any  given  value  of  6, 
then,  the  required  probabilit}-  is  ^  ~  "  .  The  i)r()I)ability  that 
6  shall  lie  between  an}'  given  value  6q  and  ^o  +  ^W  is  — . 

The  chance  that  his  first  course  shall  make  an  angle  witli  the 
river  between  6q  and  Oy^  +  dO,  and  that  he  shall  get  back,  is  " 

TT-'ie    (W       {iT--26)d6 


As  6  is  equally  likelv  to  have  any  value  between  0  and—,  the 
required  probability, 


(6)  A  floor  is  ruled  with  equidistant  straight  lines;  a  rod, 
shorter  than  the  distance  between  the  lines,  is  thrown  at  ran- 
dom on  the  floor ;  to  And  the  chance  of  its  tailing  on  one  ol"  the 
lines. 

Let  X  be  the  distance  of  the  centre  of  the  rod  frona  the  nearest 
line  ;  6  the  inclination  of  the  rod  to  a  perpendicular  to  the  jjaral- 
lels  passing  through  the  centre  of  the  rod  ;  2a  the  connnon  dis- 
tance of  the  parallels  ;  '2e  the  length  of  the  rod. 

In  order  that  the  rod  may  cross  a  line,  we  must  have 
ccos^  >  x\  the  chance  of  this  for  any  given  value  Xo  of  x  is 
. —  cos   '  -'. 

The  prol)ability  that  x  will  have  the  value  .r^  is  — .  The 
probabilit}-  required  is 

2    C     _,.r  2  c 

ttci^^o  c  -a 

This  problem  may  be  solved  I»y  another  nutliod  which  pos- 
sesses considerable  interest. 


210  INTEGRAL   CALCrLUS.  [Art.  175 

Since  all  values  of  a;  from  0  to  a,  and  all  values  of  6  from  —  ^ 
to  -  are  equall}'  probable,  the  whole  number  of  cases  that  can 
arise  may  be  represented  by 

I        I  dxdd  =  TTtt. 

The  number  of  fa^'orable  cases  will  be  represented  by 


I     idxde  =  2c. 

Hence  j)  =  — 

TTCI 

(c)  To  find  the  probability  that  the  distance  of  two  stars, 
taken  at  random  in  the  northern  hemisphere,  shall  exceed  90°. 

Let  a  be  the  latitude  of  the  first  star.  With  the  star  as  a 
pole,  describe  an  arc  of  a  great  circle,  dividing  the  hemisphere 
into  two  lunes  ;  the  probability  that  the  distance  of  the  sec- 
ond star  from  the  first  will  exceed  90°  is  the  ratio  of  the  lune 
not  containing  the  first  star  to  the  hemisphere,  and  is  equal 
to  V2^  ~"'\     The  probabilit}'  that  the  latitude  of  the  first  star 

TT 

will  be  between  a  and  a-\-da  is  the  ratio  of  the  area  of  the 
zone,  whose  bounding  circles  have  the  latitudes  a  and  a  -f-  da 
respectively,  to  the  area  of  the  hemisphere,  and  is 

2  ira^  cos  a  da  , 

=  cos  tt  tta. 

2ira^ 

Hence  p=  i  '  ^2^~  «) ^^^g ^ ^^^  _  _, 

^0  TT  TT 

(d)  A  random  straiglit  line  meets  a  closed  convex  curve  ; 
what  is  the  probability  tliat  it  will  meet  a  second  closed  convex 
curve  within  the  first? 

If  an  infinite  num1)er  of  random  linos  be  drawn  in  :i  pl:ino,  all 
directions  :iie   etpially   prol)al)le  ;    and   lines   having   any  given 


Chap.  XV.]  MEAN    VALUE    AND    TKOIiAHILITV.  ill 

direetiou  will  be  disposed  witii  ecjiial  IVeqiieiiey  all  (ner  the 
plane.  If  we  determine  a  line  by  its  distance  p  from  the  origin, 
and  by  the  angle  a  which  p  makes  with  the  axis  of  X,  we  can  get 
all  the  lines  to  be  considered  by  making  p  and  a  vary  Itetween 
suitable  limits  b}-  equal  infinitesimal  increments. 

In  our  problem,  the  whole  number  of  lines  meeting  the  exter- 
nal curve   can  be  represented  by   j    |  dpda.     If  the  origin   is 

within  the  curve,  the  limits  for  })  must  be  zero,  and  the  perpen- 
dicular distance  from  the  origin  to  a  tangent  to  tlie  curve  ;  and 
for  u  must  be  zero  and  2  it.  If  we  call  this  number  N,  we 
shall  have 

N=   ipda, 
*^ 

p  being  now  the  perpendicular  from  the  origin  to  the  tangent. 

If  we  regard  the  distance  from  a  given  i)oint  of  any  closed 
convex  curve  along  the  curve  to  the  point  of  contact  of  a  tan- 
gent, and  then  along  the  tangent  to  tlie  foot  of  the  perpendicu- 
lar let  fall  upon  it  from  the  origin,  as  a  function  of  the  a  used 
above,  its  differential  is  easih'  seen  to  be  pda.  If  we  sum  these 
differentials  from  a  =  0  to  a  =  27r,  we  shall  get  the  perimeter 
of  the  given  curve. 


Hence  N 


=  I  pda  =  Z/, 


where  Jj  is  the  perimeter  of  the  curve  in  question.  By  the  same 
reasoning,  we  can  see  that  7i,  the  numl)er  of  the  random  lines 
which  meet  the  inner  curve,  is  equal  to  ^  its  perimeter.  For  p, 
the  required  probability,  we  shall  have 


EXAMIM.F.S. 

!„^A  number  ?i  is  divided  at  raiidoin  into  two  parts  ;   find  the 
mean  value  of  their  product.  1,,^.    '!^ 

"    ()  * 


212 


INTHGUAL   CALCULUS. 


[Akt.  175. 


Find  tlio  mean  value  of  tlic  ordinates  of  a  semicircle,  siip- 
p(5sing  tlie>ierie.s  of  ordinates  taken  equidistant.  ^,^5^  ^^^^ 

(P9  Find  the  mean  value  of  the  ordinates  of  a  semicircle,  sup- 
posing the  ordinates  drawn  through  equidistant  points  on  the 
circumR'^uce.  .         2(( 

(P)    Find    the    mean    values   of  the    roots    of  the    quadratic 

ar  —  <i.r  -\-b  =  0,  the  roots  l)eing  known  to  be  real,  but  b  being 

unknown  but  positive.  ,         oa        ,  a 

^  Ans.   —  and  -. 

t;  G 

("))  Prove  that  the  mean  of  the  r.idii  vectores  of  an  ellipse,  the 
focus  being  the  origin,  is  equal  to  half  the  minor  axis  when  they 
are  drawn  at  equal  angular  intervals,  and  is  equal  to  half  the 
major  axis  when  the^-  are  drawn  so  that  the  abscissas  of  their 
extremi^es  increase  uniforml}'. 


\. 


/>^'Si 


)    Suppose  a   straight  line   divided    at  random   into   three 
parts  ;  find  the  mean  value  of  their  product.  ,         a^ 

y  "  '"•  ¥)• 

1/(7)    Find  the  mean  square  of  the  distance  of  a  point  within  a 
iiiven  square  (side  =  2r/)  from  the  centre  of  the  square. 

•^8)  A  chord  is  drawn  joining  two  points  taken  at  random  on 
a  circumference  ;  lind  tiie  mean  area  of  the  less  of  the  two  seg- 
ments iato  which  it  divides  the  circle.  .        ira-      a- 

lr[U)    Mud  the  mean  latitude  of  all  places  north  ot  the  etiuator. 

A71S.  32°. 7. 
10)  Find  tlu^  mean  distance  of  [)oiuts  within  a  sphere  from 
ven  point  of  the  surface.  Ans.    |a. 

(11)    Find  the  mean  distanc 
I    wirfiin  a  sphere. 


•f  l\v<>  points  taken  at  random 
Ans.  44  a. 


(12)   Two  points  are  taken  at  random  in  a  given  line  a  ;  find 
Ihe  chance  that  then-  distance  shall  exceed  a  given  value  c. 


.■1//.S 


<l  —  C 

a 


Chap.  XV. J  MEAN    VALUK   AND    I'i;«  M'.AHILITY.  'Jl^ 

(13)  Find  the  oliaiicc  tluit  the  distance  of  two  points  witiiin 
a  square  shall  not  exceed  a  side  of  the  s(iiiare.        ^lus.  w  —  \;'^ 

(14)  A  line  crosses  a  circle  at  random  ;  tiiid  the  chances  tiiat 

a  point,  taken  at  random  within  the  circle,  shall  be  distant  from 

the  line  by  less  than  the  radius  of  tlu'  circle.             ,         ,        2 
•^  Alls.   1 

;37r 

(15)  A  random  straight  line  crosses  a  circle  ;  find  the  chance 
that  two  points,  taken  at  random  in  the  circle,  shall  lie  on 
opposite  sides  of  the  line.  '^    -     '     «    ^         «  128  ..- 

^  45  7f  ' 

(16)  A  random  straioht  line  is  drawn  aci-oss  a  s(piare  ;   find 

the  chance  that  it  intersects  two  opposite  sides.  lo<;2 

Ans.  ^ ^ — 

TT 

(17)  Two  arrows  are  sticking  in  a  circular  target;  find  the 
chance  that  their  distance  apart  is  greater  than  the  radius. 

An.'i.   —^ — 
47r 

(18)  From  a  point  in  the  circumference  of  a  circular  field  a 
projectile  is  thrown  at  random  with  a  given  velocity  which  is 
such  that  the  diameter  of  the  field  is  equal  to  the  greatest  range 
of  the  projectile  :  find  the  chance  of  its  falling  within  the  field. 

An.s.  ^_^(V2-1). 

TT 

(ID)  On  a  table  a  series  of  ecpiidistant  parallel  lines  is  drawn, 
and  a  cube  is  thrown  at  random  on  the  table.  Supposing  that 
the  diagonal  of  the  cube  is  less  than  the  distance  between  con- 
secutive straight  lines,  find  the  chance  that  the  cube  will  rest 
without  covering  any  part  of  the  lim's. 

Ans.   1 ,  where  a  is  the  edge  of  the  cubu  anil  >•  tlie  ilis- 

tance  between  consecutive  lines. 

(20)  A  plane  area  is  ruled  with  equidistant  parallel  straight 
lines,  the  distance  between  consecutive  lines  being  r.  A  closed 
curve,  having  no  singular  points,  whose  greatest  diameter  is  less 


214  INTEGRAL   CALCULUS.  [Art.  175. 

than  c,  is  thrown  down  on  the  area.     Find  the  chance  that  the 
curve  falls  on  one  of  the  lines. 

Ans.  ■ — ,  where  /  is  the  perimeter  of  the  curve. 

TTC 

(21)  During  a  heavy  rain-storm,  a  circular  pond  is  formed  in 
a  circular  field.  If  a  man  undertakes  to  cross  the  field  in  the 
dark,  what  is  the  chance  that  he  will  walk  into  the  pond? 


ClIAP.  XVI.]  ELLIPTIC    INTEGKALS.  215 

CHAPTER    XVL 

ELLII'TIO    INTEGRALS. 

176.  In  attempting  to  solve  completely  the  prol)lem  of  the 
motion  of  a  simple  pendulum  by  the  methods  of  I.  Ciiapter 
VIII.  we  encounter  an  integral  of  great  importance  which  we 
have  not  yet  considered.  The  problem  is  closely  analogous  to 
tliat  of  the  Cycloidal  pendulum  (I.  Art.  119). 

For  the  sake  of  simplicity  we  shall  suppose  the  pendulum 
bob  to  start  from  the  lowest  point  of  its  circular  path  with  the 
initial  velocity  that  would  be  acquired  by  a  particle  falling 
freely  in  a  vacuum  through  the  distance  ?/o;  and  this  by  I.  Art. 
114  [1]  is  V2^o. 

Forming  our  diflferential  equation  of  motion  as  in  I.  Art.  118, 
but  taking  the  positive  direction  of  the  axis  of  Y  upward,  we 

ds 
Multiplying  by  2 —  and  integrating, 


ds\  .       ,  ^ 


.dt. 
or,  determining  C, 

v'  =  (~IJ= '2  (,{!,.-!,).  (2) 

If  the  starting-point  is  taken  as  the  origin,  the  equation  of 
the  circular  path  is  ar  +  y-  —  2  ay  =  0,  whence 


\dt)      2cvj-y\dtJ 


and  we  have  ,     "^       ,  '^'=  V2^o-y), 

V2  ay  —  y^  "' 


216                                 INTEGRAL  CALCULUS.  [Art.  1/ 

dt=  ^^ 


V2gr  .  N{y^-y){'lay-f) 
lutegrating,  and  determining  the  arbitrary  constant,  we  get 

t = ^  c         ^y    —  (3) 

^'2gJ^  V(yo-2/)(2ai/-/)  . 

as  the  time  required  to  reach  that  point  of  the  path  which  has 
the  ordinate  y. 


The  substitution  of  .r-  =  —  reduces  (3)  to  the  form 


^^i"T7 


dx 


^((1 -.')(. -f;.-' 


w 


where  the  integral  is  of  the  form 

C  ^^        — .  (5) 

*^»  V(l  -ar')(l-fc2a^) 

fc^  being  positive  and  less  than  unity  if  ?/u  is  less  than  2  a.  An 
examination  of  equation  (2)  will  show  that  if  this  is  true,  the 
pendulum  will  oscilhite  between  tlie  two  points  of  the  arc  which 
have  the  ordinate  ?/„. 

If  ?/o  is  greater  than  2  a,  tlie  pendulum  will  make*  complete 
revolutions.  For  this  case  the  substitution  of  x^  =  f-  in  (3) 
will  reduce  it  to 

t^aJ^f'  ""  (6) 


V<-<-ir?) 


where  the  integral  is  of  tlie  form  (o),  k-  being  positive  and  less 
than  unity. 

The  time  required  for  the  pi-ndiilum  to  reach  its  greatest 
height  —  that  is,  in  the  first  case,  the  time  of  a  half-vibration, 
and  in  the  second  case,  the  time  of  a  half-revolution  —  will 
depend  upon 

r       ^•^-    — .  (7) 


Chap.  XVI.]  ELLIPTIC    INTEGKALS.  o-jy 

177.    TIio  leiiiith  of  an  art;  of  an  Ellipse,  measured  from  the 
extremity  of  the  minor  axis,  has  been  found  to  be  (Art.  107) 


XV^fcS-''-  0) 


If  we  replace  '    by  x,  (1)  becomes 
a 


and  the  integral  is  of  the  form 


IVt 


^  •  clx,  (3) 


where  k-  is  positive  and  less  than  unity. 
The  length  of  an  Elliptic  quadrant  depends  upon  the  integral 


XV^ 


-^.dx.  (4) 


178.  It  can  be  shown  l)y  an  elaborate  investigation,  for 
which  we  have  not  room,  that  the  integral  of  any  algebraic 
function,  which  is  irrational  through  containing  under  the  sciuare 
root  sign  an  algebraic  polynomial  of  the  third  or  fourth  degree, 
can  by  suitable  transformations  be  made  to  depend  upon  one 
or  more  of  the  three  integrals 

F{k,x)        =r^=r^£^=-=,  [1] 

Jo  V(l-ar)(l -A:V) 


n{n,k,x)=  r ^^  [31 

^'   {\+  ux')  V(  1  -  .1-)  (1  -  A-»x») 

which  are  known   as  the   Elliptic  Integrals  of  the  lirst,  second, 
and  third  class  respectively. 


218  INTEGRAL   CALCULUS.  [Art.  179. 

A;,  whicli  may  alwaj'S  be  taken  positive  and  less  than  i,  is 
called  the  7nod(dus;  and  ?«,  which  may  be  taken  real,  is  called 
the  parameter  of  the  integral. 


K=  F(k,  1 )  =  f  ^^  [4] 


dx 


and  E=E{k,\) 


=xvw-'      [^^ 


are  known  as  the  Complete  Elliptic  Integrals  of  the  first  and 
second  classes. 

179.    The  substitution  of  iB  =  sin<^  in  the  Elliptic  Integrals 
reduces  them  to  the  following  simpler  forms. 

F(A-,*)=r-^!^=         =r|*.      [1] 

n  („,  ic,  </,)  =  (•* ^  =  ("^ ^'-^ 

->'«    (l+J«sin2<^)Vl-^--sin"<^     c/«(l  +  ?isin- 


[3] 

K=  c  _A^_         =  r'i^.      [4] 

Jo  Vl-Fsin^,^  Jo  A<^ 

^  =  r  Vn^"Fsin^  .d<^  =  f  A</> .  r7<^.         [f)] 

<^  is  called  the  amplitude  of  the  Elliptic  Integral,  and 
A</>  =  Vl  —  A:'sin^</)  is  called  the  delta  of  <;^,  or  more  simply, 
delta  <^,  and  is  regarded  as  a  new  trigonometric  function  :  it  is 
always  taken  with  the  positive  sign,  and  has  an  analogy  with 
cos«^. 

For  a  given  value  of  A-,  A<^  is  easily  seen  to  be  a  periodic 
function  of  <f>  having  the  period  tt.    It  has  its  maximum  value  1 


Ck.vp.  XVI.]  ELLII'TIC    INTKOKALS.  219 

when  ^  =  0  and  when  <^  =  tt,  and  its  niiuinuira  vahie  \'  1  —  k\ 
which  is  usually  represented  by  A'  and  called  the  complemenUu-y 

modnhis,  when  <i  =  -  ;  and  A(-  +  (M  =  A['^  — a 


Lauden's   TnoisfornKdiun. 

180.  The  approximate  numerical  value  of  an  Elliptic  Integral 
of  the  first  class,  when  k  and  <^  are  given,  is  easily  computed 
by  the  aid  of  two  valuable  reduction  formulas  due  to  Landen. 


If  in  Fik,cl>)=r-—^ 

Jo  ^J^  _  z-2 , 


Vl  —  k-sm-cf) 
we  replace  <^  by  <^i,  </>,  and  </>  l)eing  connecicd  by  the  relation 

,       ,          sin2<i,  >-^ 

tand)  = ^-^—,  (1) 

^'-^cos2<^l  '^ 

which  is  easily  transformable  into  either  of  the  following : 

^•sin<^  =  sin  (2  ^1  —  ^),  (2) 

tau(<^-c^,)  =  f^-f  tanc^,,  (3) 

I               ^             reduces  to    — - —  |    ' — 
^0   Vl-^-^sin^<^  1+^v^o       L i^sin^c^, 

which  is  also  an  Elliptic  Integral  of  the  first  class,  but  has  a 
different  modulus  and  a  different  amplitude  from  those  of  the 
given  integral. 

The  steps  of  the  process  are  as  follows : 

From  (1)  we  easily  find 

8in2  2<^, 


siu-c^ 


1  +^-^+2A:c()s2</), 


,  n —  I      -^r-r  1  +  Ar  cos  2  <^ 

whence  VI  —  Arsu]-^  =       — — :• 

Vl  4-A--  +  2;.cos2</), 


220 


INTEGRAL   CALCULUS. 


[Art.  180. 


Differentiating  (1),  we  get 


2j  7.       2(1  -^-kcos2<b^,, 
(^•  + cos  2^1/ 


I  lilt  from  (1), 

hence 

cl<f>  _ 


secV 


1  +A-  +  2A-cos2<^i. 
(A;  +  cos2<^i)^      ' 


2(l+keos2<f>o 
1  +  A;-  +  2  k  cos  2  <f>, 


2d<^i 


2d<f>, 


Vl-A;^sin>      Vl +A;2+ 2fc  cos2<^i      Vl  +  A,-^+ 2A;-4A;siu"'^0, 


rfc^i 


\-i-k 


x(l ^,sinVi 


<f)i  =  0     wlien     <^  =  0, 

hence       I  ^    = I 

Jo    Vl  -A-- silled,      1  +  k  Jo 


r7<^i 


4A-       .   , 
\  1 ^.sur 

\      (1+A-y^ 


<^i 


and 


F{k,c}>)  =  -^    F{k„cj>,). 


where 

^•i 

_  2VA-^ 
1  +  A-' 

and 

sin(2<^ 

-</>) 

=  A  siii<^ 

A-,  is 

less 

th 

.111 

1 

and  < 

greater  th 

[4J 


;   for  <  1  reduces 

14- A; 

to    0<(1  — Va-)-,    which    is   obviousl}'   true,    and  >  A; 

1+A: 

reduces  to  4  >  A(l  +A-)-,  which  is  true,  since  A-  is  less  than  1. 
If  ^  is  not  greater  than  ^,  and  the  smallest  value  of  <^i  con- 
sistent with  the  relation  sin  (2<^,  —  ^)  =  A' siu<^  is  taken, 
0  <  <^i  <  c^.  Ilonce  (4)  is  a  reduction  formula  by  which  we 
can  raise  the  nioduUis  and  lower  the  amplitude  of  our  given 
function. 


CllAP.  XVI.]  ELLIPTIC   intk(;i:als. 

By  applying  the  fonnuljv  (4)  n  times,  we  get 


221 


or,  since 


l+A-     1-fA-,     1+A\ 
= — -,  etc., 


-l-A-„  , 


nK,<i>n)\ 


F{k,  <f>)  =  A-,.  ^^•^^•^-K  .  ^(^.  _^  ^^ )  ^ 


where 
A- 


^  2VA;  I 
1  +  K  I 


and  sin  ( 2  <^^,  —  <^^,_i )  =  A-^,  i  sin  <^^,_, . 


D'] 


If  we  suppose  ?i  in  (5)  to  be  indefmitely  increased,  we  shall 
have  rA-,,]  =  1  ;  for  if  we  form  the  series 

(l-A-)  +  (l-A-04-(l-A-,)  +  -"+(l-Av)-f--, 
we  shall  have 

1  -  Av      1  -  A-,       1  -  k/    1  +  va-^  1  +  k; 

which  is  always  less  than  unity  ;  hence  the  terms  in  the  seric-a 

must  decrease  indefinitely  asj)  is  increased  and    _      [1  — A"„]  =  0. 

Since,  as  we  have  seen  above,  <^„  continually  diminishes  as  n 
increases,  but  does  not  reach  the  value  zero,  it  nuist  have  some 
limiting  value  <I>.     Hence 

^^-"^  ^^    Vl  -sin-<^ 

=  1    sec  <l>(I(f>  =  log  tan  ^  +  -    ; 

and  F(k,  0)  =  log  tanf"^  +  |1* /5SI^.  [6] 

Formulas  [/)]  and  [6]  lend  themselves  very  readily  to  numer- 
ical computation. 


222 


INTEGRAL   CALCULUS. 


[Art.  181. 


181.  Formula  [4J,  Art.  180,  may  be  used  to  decrease  the 
modulus  and  increase  the  amplitude  of  a  given  Ellii)tic  Integral. 
Interchanging  the  subscripts,  and  using  (3)  Art.  180  instead  of 
(2)  Art.  180,  we  have 


F{k,<f>)='^-thF(k„cf>,), 


where 
and 


Jc, 


1-vr 


1  4.Vl-A;^ 
tan(<^,  —  (^)  =  Vl  —  k-tiin4 

By  repeated  application  of  [1]  we  get 


[1] 


i^(A:,<^)  =  (l+A-,)(l+A-,)...(H-/0 


^(^•«,  </>..) 


when 


and 


tan(<^^  -  <^^^i)=  Vl  -  A--^  1  tan<^^  1. 


[^] 


limit 


It  is  easily  shown,  as  in  Art.  180,  that     _      [A'„]  =  0,  and 

limit  /** 

consequently  that    _     F{k,„  <^„)  =1    d(j!)  =  <i>,  where  4>  is  the 

n 00  ^y 

limiting  value  approached  by  (f>  as  n  is' increased. 

If  <^  =  ^,  we  get  from  [2],  c/>i  =  tt,  </>.  =  2  7r,  ...<^„  =  2"-^; 


hence 


K=f(j.;  ^)  =  ^  ( I  +  h)  (1  +  /.■.)  (1  +  A-.) 


[3] 


Formulas  [2]  and  [3],  like  foiniulas  [;">]  and  [6]  of  Art. 
180,  lend  themselves  readily  to  computation. 

"NVitli  a  large  modulus,  it  is  generally  best  to  use  [5]  and  [6] 
of  Art.  180;  with  a  small  modulus,  [2]  or  [3]  of  the  present 
article  will  gonorally  work  more  rapidly. 

We  give  in  the  next  article  the  whole  work  of  computing  the 

Elliptic  Integral    /'Y-^,  -  )   by  each  of  the  two  methods,  and 


Chap.  XVI.]  ELLIPTIC   INTEGRALS.  223 

of   computing    k(-—\  =  f(—^,'^A    by   tlie    second   method, 
employing  five-place  logaritlims. 

182.  Ff^,-\     Metiiou  of  Art.  180. 

fc=  0.70712  log^  =  ;».s4949 

1+^-=  1.70712         log  (1+ A;)  =0.2^226 

log  V^  =  9.92474 

log2  =  0.30103 

colog(l+^')=  9-76774 

logA-i  =  9.99351 

A"i  =  0.98518  logA:i=  9.99351 

l-i-A;i=  1.98518         log(l +Ai)  =  0.29780 

log  VA?i=  9.99676 

log2  =  0.30103 

colog(l  4- A-i)  =  9.70220 


logA-,= 

9.99999 

h,= 

1 

logic  = 

9.84949 

log  sin  ^  = 

9.84949 

2*^1 -</,)  = 

9.09898 

2</)i  — (/)  = 

30°    0' 

3 

2*^1  = 

75°    0' 

3 

•^1  = 

37°  30' 

2' 

log  A;,  = 

9.99351 

log  sin  ^,  = 

9.78445 

log  sin  (2(/)j  — <^,)=  9.77796 


224 


INTEGRAL,    CALCULUS. 


[Akt.  lf.2 


2<^2-«^,  =  36°  51'     3" 


2^2 


74°  21' 


$  =  <^2  =  37°  10'  32" 
^$  +  !i:=63°  35'  16" 

log  tan  /^^  +  ^d.^  =  0.30393 


log  log  tan  (  -  +  i  <I> 


log  Va'i  =  9.99676 
cologV^  =0.07526 
9.48277 


log-F' 


colog/x  =  0.36222 

V2 


2     4 


=  9.91701 


i?'/^:Zl  e"\  =  0.82605 

V  2  uy 

/ui  =  0.43429    is    the   modulus   of   the    common    system    of 
logarithms. 


rf^,"^ 


Method  of  Art.  181. 


VI  -/.•-  =  /.'  =0.70712 

1-/,'  =0.29288 

1  +/.'  =  1.70712 

/.•,  =  0.1 7157 

1_7,-,  =0.82843 
1  +/.-,  =  1.17157 


Z-,'=  0.98520 

1  -/.,'=  0.01480 

1  +/./=  1.98520 

A:2  =  0.00746 


,()g(l  -/,-')  =  9.46669 

c'ol()g(l  +  A-')  =,9.76774 

logA-,  =  9.23443 

log(l  -/.•,)  =  !). 91826 

lou(l  +  A:,)  =  0.06878 

log  A-,'- =  9.98704 

log  7.-,'   =9.^9352 

Iog(l  -A/)  =  8.1 7026 

colog  ( 1  +  AV)  =9.70220 

logA-2=  7.87246 


Chap.  XVI.]  KLMI'TIC    INTEGRALS.  225 

1  _  h,  =  0.992 ')4  log  ( 1  -  A-,,)  =  9.9907") 

1  +  k.,  =  1 .00746  log  ( 1  +  /.-,.)  =  0.003-23 

log  A-,'- =  9.99998 
log  A/   =9.99999 


logy 

logA-'  =  9.84949 
log  tan  <^  =  0.00000 

log  tan  ((/>!  —  <^)  =  9.84949 

<f>^-cf>  =  Sij°  15'  53" 
</>,  =  80°  15'  53" 

logA.V  =  9.99352 
log  tan  ^1  =  0.76557 


log  tan  (c^2  -  <^,)  =  0.75909 

<^,  _<^i=    80°     7'  17'' 
<f>.,=  160°  2:3'  10" 

tan  (<^3  —  <^o)=tan^2 

4)  =  <^3  =  2 <^,  =  ;520°  46'  20" 

1$=    40°     5'  48" 

= 144348" 
TT  =  648000" 

colog7r"=4.18842 
log  77=0.49715 


log(i*)   =i 


9.84499 


226  INTEGKAL   CALCULUS.  [Aur.  183 

log  (1 +A-i)  =  0.06878 
log(l  4- A2)  =  0.00323 


log('|3<I>^  =  9.84499 

logi?'(^^,^')  =  9.'J1700 


F{^,'-^\  =  0.82G05 


Vi 


For  F[~,~\  we  have  by  (3),  Art.  181, 

log(l  +A:i)=  0.06878 

log(l  +/t2)  =  0.00323 

log7r  =  0.49715 

colog  2  =  9.69897 

logi^f— ,  'M  =  0.26813 


i 


V2 


1.8541 
2     2, 


183.  Landen's  Transformation  can  also  be  applied  to  the 
computation  of  Elliptic  Integrals  of  the  second  class,  but  the 
task  is  a  more  dillicult  one ;  we  shall,  however,  give  a  brief 
sketch  of  the  method  ;  and  in  so  doing  we  shall  apply  it  to  a 
more  general  form 

-^'J   VI  — A''sin-> 
of  which  S(fc,  <^)  is  a  special  case. 
From  Art.  180  we  have 

.  /i 'n~^  ^n  1  +  A;  cos  2  <A. 

Vl  +  Jc-  +  2k  cos  2  <f}i 
cos<^= ''^  +  eos^, 


Vl  +k-  +  2kcos2<f)i 
2(l+kcos2<f>,) 
^      l  +  A;^+2Acos2  0i   ^ 


Chai-.  XVr.]  ELLIPTIC    INTKCltALS.  227 

Hence  Vl  —  k"  sin-  (f>  +  kcos  ^  =  Vl  4-  k'  +  '2k  cos  2  <^,. 


•^^'Vl  —k'-sm-cji 


A.  l-(i-k-i 


^-2sin->     ^"        Vl-A-sin->    J 


=r 


a  + 


.Vl-A-sin2<^     /c" 


-#.vr= 


A:^sin-<^ 


rf<^, 


and 


G(k,  <^)-^sin. 
A' 


^0    [_V1  — /i-sin-^      A. 


Snbstitnting  </)i  for  <^,  this  becomes 


0'(A-,  c^) 


k  Ji>     Vl+A;-  +  i 


cos  2  <f)i 


2  it  cos  20, 


f?<^i 


1  +  A- Jo 


a_|  +  Msin^^. 


d</,,. 


Hence  C? (A-,  cf>)  =  ^sin c^  +  -"     <7,(A„  c^,),  [2] 

where 

A-,  =  |^;,   sin(2  0i-c^)  =  A-sinc^,  a.=  --f,   y8i=\^-         [••^] 

Formnlas  [2]  and  [.")]  enal)le  ns  to  niaUe  onr  given  fnnction 
depend  npon  one  of  tlie  same  form,  bnt  liaving  a  greater 
modulus  and  a  less  amplitude.  A  repeated  use  of  [2],  together 
with  the  reductions  emi)loycd  in  Art.  180,  gives  us 


228  INTEGRAL   CALCULUS.  [AllT.  183. 


G(k.  <^)  =  |sin<^  +  Asin<^,-f-  ^1  •  /8osin«^j 


+  fcn  ^MiJ^>  .  6^,.  ( A-„,  <^„) ,  [4] 


where  /8^=        "  ^ 


and  /3A   I  2  J    2^    ^        ^        2^-^       V 

A'  \^        ki      /I'l  A;^  A.'i  ^^  A-'a  •  •  •  A^  ly 

Just  as  in  Art.  180  k,^  rapidly  approaches  1  as  n  is  increased  ; 
the  limiting  vahie  of  6r„(A-„,  <^„)  is  then 


limit  G,XK,  </>„)=  (•"""■> +  /^''^'">d<^ 
c/«  cos  d> 


COS(f> 

=  (a„  +  /3„)  log  tan  (j  + 1^)  -  i8„  sin  </»,.     [6] 


By  Art.  180,  [5]  and  [G], 
limit 


lit  k^yj^^^h:^  log  tanj"^  +  ^"1  =  F{k,  4>). 
[4]  can  thus  be  written 
Gik,^)  =  F{k,^)L-^J\+l+   "" 

flJi  ATg ...  A;„  ]      k'l  Ao  A3 ...  A'„  j/ J 


^\        A-i      A,As 


+  ^   sin  <l>  +  ~  sin  </>,  +  — ^  sin  <f>2  +— ^^  sin  <^  + 
^L  VA-  VAA-,  ^/kk,k2 

2'i-i  2"  "I 

,  sin (^„  1  -    ,_.,  sin <^„  .  [7] 

VfcA-i...A;„  2  VaAv-Vi  J 


CllAP.  XVI.]  ELLII'TIC    INTKGUALS.  22'J 

If  a  =  1,  and  fS  =  —  A-,   [7]  reduces  to 


A-'i  A*2  •  •  •  A^rj- 1         A'l  a'2  •  •  •  a',j  _ 


—  A-   sin  ^  + -^  sin  c^i  H — ; — ^sin(^2  +  "* 


2     .  2- 

—  sm(^i  +  ---^ 

Va-  Va-a-, 


siu  d),. 


VA-A'i...A-,._,,  ^kki...k„ 

_2VA~ 


in<^«i  [«] 


where     A;^  =  "      /^' ,  and  siu  (2  <^;,  -  <^,,  ,)  =  ^V  i siu  <^p  i-     [9] 
i  +  A-;,  1 

By  Formulas  [8]  and  [9]  an  Elliptic  Integral  of  the  Second 
Class  may  be  computed  without  ditliculty. 


184.  Formula  [2],  Art.  183,  may  be  used  to  decrease  the 
modulus  and  increase  the  amplitude  of  an  Elliptic  Integral. 
Interchanging  the  subscripts,  we  have 

G  (A-,  4>)  =  ^-^[g,  (A-„  <^0  - 1  sin  <^,1 ; 

or,  since  f  =  f .  (Art.  183  [3]), 

A"i      2 


G 

where 


(A-,  <^)=  ^-4^'|  G,  (A„  c^,)  -  /^  sinc/,,1  [1] 


fc,  =  f7-'{L^,  tan(<^,-<^)  =  Vr-A-='tan<^,«,  =  »+^,/?,  =  ^''/. 

[2] 


230  INTEGRAL   CALCULUS.  [Akt.  184. 

A  repeated  use  of  [1]  gives 

G-(A,  </))  =  --—  .  -y—  •  — — Or„(A,.,  <^„) 

,  1  +  A,     1  +  A'.,      1  +  /'•„  o        •     ,  1  roT 

+  -~-^  •  -^-  •  •  •  -^  A.  1  sm  0  J,  [3] 

where  y3^,  = /^MAill^, 

and      s=a  +  i/3^  +  |  +  ^^  +  ^|f^'  +  --+^''^'y;\M- 
Just  as  in  Art.  181   limit  A-„  =  0,  therefore  limit  ^,,  =  0  and 

X<t>n 

By  Art.  181,  [2],      ^-±h  .  i±h...l±h>^^^  =  F{k,  <p)  . 

By  Art.  180,  [5],      l±ij  =  2^,    ^-±^  =  2^,  etc. 
Hence  [3]  becomes 
f?  (A,  c^)=F(A-,  </,)!"«  + 4/3(^1 +|+^^+^+-)1 

-  ^  I  — '  sm  (^1  +  — ^-  smc^o  +  — ^'  sm  «^3  +  ••.  J-       [4] 
If  u=  1   :md  (i  =  —k-,  [4]  reduces  to 


Chap.  XVI.]  ELLirTIC   INTEGRALS.  2:11 

+  fc[^'  sin  <^,  +  ^'  sin  <t>,  4-  ^^^^'  siu  «^,  +  •  •  •],        [a] 
where 


i-^i-kU 


K  = 'A  -'^"^1  tan  (<^^  -  <^^,  i)  =  Vl  -  A;  i  •  tan  <j>^  ,. 

1  +  Vl  -  A-;  1 

[C] 

We  have  seen  in  Art.  181  that  if  ^  ='^,  d,  =9p  i_ 

Therefore,   for  a  complete   Elli[)tic  Integral    of   the    second 
class  we  have 

Formnlas  [o]  and  [7]  are  admirably  adapted  to  compntation. 

We    give    in    the    next    article    the    work    of    computing 

E[ — , -)   by  each  of  the  methods  jnst  aiven,  and  of  com- 

puting    ^(~o"'o)  ^^y   the    second   method;    using,    as    far   as 
possible,  tlie  values  already  employed  or  oljtauied  in  Art.  I.s2. 

185.  Ef'^,-\     Mf.tiioi)  of  Art.  183. 

Here,  as  we  have  seen  iu  Art.  182,  if  we  carry  the  work  only 
to  five  decimal  places,  k.,=  1,  and  our  working  formula  will  be 

E{k,cj>)  =  F(k,<f>)\']+kfl-pj] 

r  '*  2-        "I 

—  k\  sine/)  -(-— ^siu(/),  —  ■ siuf/j.  • 

L  V^  Va-a-,         i 


232 


INTEGRAL   CALCULUS. 


[AuT.  185. 


log  2 

logA- 

colog  A'l 


0.30103 
9.84949 

0.00G49 


log(^ 


0.15701 

^=1.43553 
A:, 

1+A:=  1.70712 

1  +  A- -^■'\  =  9.43391 

9  ;• 
1  + A; -  —  =  0.27159 

ogF/'^,-^  =  9.91 701 
9.35092 

'&■ 

i)C+'-f)='-''''''' 

log  A;  =9.84949 

log  sin</)  =  9.84949 

9.69898 

ksmct>  =  0.6 

log2  =  0.30103 

logV^'  =  9.92474 

log  sin  (^1  =  9.78445 

0.01022 

—  sin  </>x=  1.0233 
VA; 

log  22  =0.60206 

log  V^=  9.92474 
colog  V77i  =  0.00.324 

log  sin  «/>2=  9.78122 

2 

-   -  sin  <f> 

VA: 

^'^  sin<^o- 2.0477 
VA;A-i 

0.31126 

-  JcfsUKf)  -\- 

^  sin  <^.,^  =  0.5239 

V/Ta^       7 

<f. 

4)C-^^-f)  =  '-^^^''' 

£(^^^)  =  0.74825 

Chap.  XVl.J 


JiLLll'TlU   INTEUllALS. 


233 


E 


my 


Method  of  Art.  184. 


E{k,4>). 


:i^(^,<A)[l-f(H-|+'f)] 


+  k(^^  sin  <^i  +  ^^^^'  siu  </>; 


\  2 

log^-,  =  9.23443 

logA-2=7.'^7246 

Colog4  =  9.39794 

6.5U483 


2- 


^  =  0.00032 


^=0.08578 
2 


l+^  +  M?=i.08610 


1(1+1  +  ^-^'')  =  0-^71525 
logO.728475  =  9.86241.5     1  _  |Yl  +  ^  +  ^^^  =  0.72847o 


logFf^^,j)  =  0.91 700 


V-i 


9.77941.0     FfJir,!!-]  (0.72847.5)=  0.GU17H 


IogA-=  9.84949 

logV^,  =  9.61722 

colog2  =  9.69897 

log  sine/.,  =  9.99370 

9.1.5938 


A;A- 


<>,  =  0.14434 


234  INTEGRAL  CALCULUS.  [Art.  185. 

logfc  =  9.84949 
logV5  =  9.61722 
log  VA;2  =  8.93623 
colog4  =  9.39794 

log  sin  </)2  =  9.52592  

7.32680  ^^^2  si„^^  3,  0.00212 

fe  f:^  sin  <^,  +  %^-  sin  4>^  =  0. 14646 

i?'/'2^, '^Vo. 728475)=  0.60178 


Ef^, -\  =  0.7482i 


Ef^,'!^\     Method  of  Art.  184. 

:^'2)=^t^'i)[^-iO^^'^o} 

l--fl  +h^hh\  =  0.728475  logO.728475  =  9.862415 

2  V        -^        '^'  / 

WF/^— .-^  =  0.26813 


:<fi)  =  o.. 


e(^,  ''^  =  1.3507  logEf^,  "  ]  =  0.13054 


■>'2J  °     V  2    2 


CllAi'.  XVI.]  ELLTPTrc   INTEGRALS.  235 

186.  An  Elliptic  Integral  of  the  tirst  or  second  class,  whose 
amplitude  is  greater  than  -,  tan  he  made  to  depend  upon  one 
whose  amplitude  is  less  than  '^,  and  upon  the  corresponding 
Complete  Elliptic  Integral. 

AVe  have 
F{7c,.)=  f^=  r^+  f^  =  A'+  i'\  bv  [4],  Art.  179. 

In  r^  let  <^  =  TT  -  i/. ; 


then  d(}>  =  —dij/  and  A<^  =  Vl  —  Jc'  sin^<^  =  Vl  —  A"-  sin'-'i//  =  Ai/', 

n  0  0  !: 

and  we  have      r  d^  ^  _  T  #  ^  _  rO^  _  r-d^  ^  j^ 

2  2  2 

Hence       ^(.,  .)=p|^=  2/f.  [1] 


F{k,n^  +  p)=:J'^'^ 


A<^ 


^fr?<^    /-j^    rd±    ^     r±<t>,„,  f# 

./  A(^      J  A<^      J  A(^  J  A<^  J  A</> 

"  TT  in-  pn  n" 

(p+nf 

In  r^  let  <i>=i>TT  +  ip\  then  (/<^  =  (/<//,  and  A«/)  =  A./., 
J  A<^ 

pTT 

and  we  have       "H^  =  fW  ^  C'^4>  ^  ,  A^ 


236  INTEGRAL  CALCULUS.  [Aut.  186. 

The  substitution  of  if/  for  <^  —  ?i7r  iu  j  —  gives  us 

nT+p  p  p 

J  A(f>     J  A(//     J  Ad) 
Therefore         i^(A;,  mr  +  p)  =  2?i/ir+  F(^-,  p).  [2] 

In  like  mauner  it  can  be  proved  that 

F{k,ii7r-p)  =  -2nK-F{k,  p),  [3] 

E  (A-,  7i7r  +  p)  =  2  n^  +  E  (A-,  p) ,  [4] 

E{k,n7r-p)=2nE-E{l\  p),  [5] 

where  E  =  E(k,-\  is  the  complete   Elliptic   Integral   of   the 
second  class. 

A  table  giving  the  values  of  the  Elliptic  Integrals  of  the 
first  and  second  classes  for  values  of  the  amplitude  between 
0  and  -  is,  then,  a  complete  table. 

Such  a  table,  carried  out  to  ten  decimal  places,  is  given  by 
Legendre  in  his  "  Traite  des  Fonctions  EUiptiques."  We  give 
in  the  next  article  a  small  three-place  table. 

It  must  be  noted  that  the  first  column  gives  F(0,  <f>)   and 

£(0,  <^),  that  is,  I    d<f}  =  <t};    and  that  the  last  column  gives 
F(l,  <^)  and  ^(1,  </>),  that  is,  log  tan /''' + -Vnd  sin<^. 

Tiie  complete  Elliptic  Integrals, 


/f=F(/.-.;     „n,lA'=A7/,-, 


are  given  in  the  last  line  of  eacii  table. 


CiiAi'.  XVI.] 


ELLIPTIC    INTEGRALS. 


237 


t>i  4-  -4 


'p'pp 

C\  (^  in 


•-q  C^  '-" 

O   H-   tsJ 

O  NJ  tJi 


ooo 

4-  bo  N3 

U)4-  Cs 


pop 

•^J  CO  p 
Cn  ^I  P 


OOO 

^icoo 

'-n  -a  O 


a,  ^ 
5' II 

ClO 


3    II 


-^  U>  to 

U>4^  ON 


OOO     ooo 


-J  ^  ^I        O  O  tsi 


pc>p 

4-  u>  to 
_  -  4-  VI  On 
OsO       Oi— OJ 


pPP 
Ln  ^1  p 


i-OO 
O  vO  CO 
oI^4^ 


poo     PPP 

^ODp 

VT^  P 


K-  N)  U) 


2.  i>r- 
9  II 


coo 


loco 

'COCO 


oi^i 


00^  < 

-f'  o< 

CO  U\  ( 


OvDCO 
U)  to  to 

-t^COON 


'OO 


OO 

4^  00 


PPP 

^  Cv  '^ 
to  U>  OJ 

•^  OON 


-I  c^  •-" 

Oj  Oj  Oj 


OOO 


PPP 

4»-  U)  to 
4»  i^  On 

U>  Ui  U> 


000 
4>.  U>  to 
4-!^.  ON 


000 


000 

t-op 
*J  CO  p 


U)P  I— 
<^>  CT>  "^ 


VI  H-  t/3    I     ;;n  VI  -i- 

4.,  H-  1— •    I     W  U)  O 


oo< 

^o< 
^ico( 


B    II 
!feO 


£9 
oO 


5r 


238 


INTEGRAL   CALCULUS. 


[Art.  187. 


II  e 

OO  o 


odd 


OO  — 

odd 


O  t^'t- 

8  CO  t^ 
O- 

ddo 


Q  CO  l^ 

OO  — 

odd 


0\  N  ro 
CO  ro  ■*- 


o  t^  •^ 


o  o  — 
t^  t^  CO 

odd 


d><o<5 


t^  o  CO 

OvCO 
O  On  O 


Is: 


VOOOON 
On  O  0^ 

ddd 


0«^  'J- 

ocoi^ 

OO  — 


C^O  ro  •^ 

ddd 


o^ 


lO  GO  ON 
—  On  t^ 

LO  lO  VO 

odd 


o  vc?o 

<M  O  ON 
IT)  vO  vO 

ddd 


in  ro  — 

t^  CO  o^ 
ddd 


On  fO  "1 

d  — — 


II  c 


OCO  l-^ 

OO  — 

ddd 


ddd 


(M  O^  VD 
O  -t-  ro 
CN3  ro  't- 


NONVO 

NO  ■^  CO 
CNlrO-^ 

ddd 


cvj  O  O 
li-.  so  vD 

ddd 


fO  On  vO 
CNJ  O  On 

u->  ONO 

ddd 


i^oco 

(M  —  ON 

W-j  O  NO 

ddd 


ON-:^CO 
I^NO   1- 

t^  CO  C7N 


CO  NO  "^ 

ddd 


>/^  CN]  ON 

CO  t^  >o 

I^COOn 

ddd 


CO  t^NO 

t^OOON 

ddd 


ro  t^  O    I   ro 


—  NO( 
tT  cm  . 
O  — < 


\0  <M  CN 
O  —  c^ 


t--rt-CSl 

SrO  CM 


NO  <^  I 
OOn( 


Chap.  XVI.]  ELLlPnC   INTKGKALS. 


Addition  Formulas. 

188.  The  Elliptic  Integrals,  F(k\  x)  and  E  (k,  x),  may  be 
regarded  as  new  functions  of  x,  delined  by  the  aid  of  defmite 
integrals  ;  namely, 

see  Art.  178,  [1]  and  [2]. 

We  have  seen  how  we  may  compute  their  values  to  any 
required  degree  of  api)roximation  when  k  and  x  are  given. 
It  remains  to  study  their  properties. 

We  are  familiar  with  other  and  much  simpler  functions  which 
may  be  defined  as  definite  integrals,  and  whose  most  important 
properties  can  be  deduced  from  these  definitions. 

— ,    sin'^'x    as 
x 

X' —  .  tan  ^x  as  |    — '- — -,  and  the  theorv  of  these  func- 

Vl  -.r'  Jo  \  ^x" 

tions  may  be  based  upon  these  definitions.     For  instance,  the 

fundamental  property  of  the  logarithm  is  expressed  by  what  is 

called  the  addition  formula, 

log  X  -\-  logv  =  log  (xy) , 

and  the  whole  theory  of  logarithms  may  be  based  on  this 
property  ;  and  there  are  adtlition  formulas  for  the  other  func 
tions  defined  above  ;  namelv. 


sin~'x  +  sin  'y  =  sin  '  (.rVl  —  //-  +  //Vl  —  x') , 

tan  'x  +  tan'y  =  iiin-^(^-^-^\ 
\\  -xy/ 


240  INTEGRAL   CALCULUS.  [ART.  188. 

These  three  important  formulas  are  usually  obtained  by  more 
or  less  elaborate  methods  involving  the  theory  of  the  functions 
which  are  the  inverse  or  anti-functions  of  the  log.T,  the  sin^a;, 
and  the  tan^a;,  that  is,  of  e'',  sin  a:,  and  tan.T;  but  they  may 
be  obtained  without  difficulty  from  the  definitions  of  log  a;, 
sin  'a;,  and  tan"'x,  as  definite  integrals. 

Take  first  log.v=  |    — • 

Jl      X 

Let  us  determine  y  in  terms  of  x,  so  that 

log.T  +  log?/  =  logc,  (1) 

where  c  is  a  given  constant. 

Since  log?/=  |    -^, 

'  ■      -^'    y 

if  we  differentiate  (1),  we  have 

^  +  ^  =  0, 

X        y 

or  ydx  +  xdy  =  0.  (2) 

Integrate  (2),  and  we  get 

Cyfjx  +  Cxdy  =  C.  (3) 

Simplify  the  first  member  of  (3)  by  integration  by  parts; 

^y  ~  \  ^(^y  +  ^'11  —  I  ?/<''^'  =  d 

or  2xy—i  {xdy  +  7jdx)  =  C. 

Reducing  by  the  aid  of  (3),       '2xy=  C, 
or  xy  =  Ci,  (4) 


Chap.  XVI.]  ELLIPTIC    IN'TEOItALS.  241 

where  C,  is  an  umletermined  constant.     To  determine  C„  let 
x=l  in  (4),  and  we  have  y  =  Ci  when  x=l  ;  let  x=  1  in  (1), 

—  =  0,   lo<5  ij  =  log f,   and    y  =  c,    when   x=l. 

Therefore   Ci  =  t',    and    xy  =  c.       Consequently    ?/=^    is   tiie 

a; 
required  value  of  y,  and  we  have  (1) 

log.x'  -f-  log-  =  logc. 

X 

We  can  express  this  relation  more  neatly-  b\^  replacing  c  by 
its  value  xy,  and  thus  we  reach  our  required  addition  formula 

\ogx  +  logy  =  \og{xy).  [5] 

189.    The  addition  formula  for  the  sin~^  can  be  deduced  in 
exactly  the  same  way.     We  wish  to  determine  y  so  that 

sin^.r +  sin^'y  =  sin^^c.  (1) 

We  have  sin  !;«=  C\^^      .  8in^'y=  f—^—. 

Jo     yj\—x^  Jo    Vl— 2/"^ 

Differentiate  (1). 

-^^+      ^=0,  (2) 

yfY^^      \l\-y- 

or  Vl  —y^  •  dx  +  Vl  —x-  •  dy  =  0, 

fVl  -  y-  '  dx  -f  fVT-^'  .  dy  =  C. 
Integrate  by  parts,  and 

J       VVT^r'      Vl^V 
or,  reducing  by  (2), 

X  Vr^^  +  y  Vn^'  =  C.  (3) 


242  INTEGRAL  CALCULUS.  [AuT.  189 

To  determine  C,  we  have  from  (3)  y=C  when  iK  =  0,  and 

—  =0,  when 

x  =  0.    Hence  C  =  c,  and  x  vT^  V'  +y'^^  —  .r  =  c,  and,  finally , 
sin  ^^•^-siu  ^y  =  sin"'  (xVl  —y'  +  y^^  —a^).  [4]. 

To  get  an  addition  foimula  for  the  tan  \  a  slight  device  is 
required,  that  of  dividing  the  ditiereutial  equation  correspond- 
ing to  (2)  by  1  —^y^. 

As  before,  let 

tan~^a;+ tan~^?/  =  tan~^c,  (5) 

ilx 


where  tan  ^x=  j    ;; — ^, 

.70  1  -\-xr 

and  tan  ^2/=  rVV^' 

c/o   1  +  y 


dx  dy     ^  Q 

(1  +  f)  Ox  +  ( 1  +  .r-)  (?2/  =  0.  (6) 

Divide  by  1  —x-y^  and  integrate. 

J  I—  7?y^  J  \—  xrxf 

Integrate  by  parts.     We  have 

1-ar?/-      (l-rr-?/-)" 


1  -  .^•2/      (1  -  ar y-) 

and 

a;  .    ^  +  ?/'   -I- ,/ .  1±-'L  ^^•  +  ?/. 
1  _  af'y-      •      1  -  xhf     1  -  a;?/ 


CilAP.  XVI. J  ELLIPTIC    INTEClllALS.  243 

Hence 

Therefore,  bv  (6),  l±JL=c.  (7) 

1  —  .I'll  ^ 


To  determine  C,  we  have  from  (7)   _?/  =  Cwhen  .r=0,  and 

J«0      fl  J. 
— - —  =  0  when 
0  1  —  or^ 
a;  =  0. 

Hence  C=c,  and  ^^^^z=c, 

1  -  x;i 

and,  finally,  tan  ^r  -f  tan  '  y  =  tan  ' (^'^■'\  fg] 


190.    To   got   an    addition   formula   for  F{li,  x),   as  before 
let  F{k,x)  +  F{k,y)=F{k,c),  (1) 

where  Fik.  x)=  C  '^^ 

-^0  V(l  -ar')(l-A:-.r) 

and  F( 


^0  V(i-yo(i 


^"/) 


'^"^  -A-  ^y  =0.  (2) 

V(  1  -  ar')  ( 1  -  A-^x-^)      V  ( 1  -  2/'0  ( 1  -  ^-^ r ) 


or 


V(l  -r/2)(l  -^-^y^^  .d:K-F  V(l  -x'){\-1cx')  •  (l!J  =  0.     (3) 
Divide  by  1  —k"jry-  and  integrate. 


244  INTEGRAL  CALCULUS.  [Akt.  190. 

Integrate  by  parts.     We  have 

-(H-^'-')(l+A^r^2/2)]  ^y 


V(i-r)(i-A:V) 


V(l-ar)(l-^•-ar^) 
+  2  A-^a;?/  V(l  -  a^)(l  -  Ar^or')  •  dy j . 
Hence 

a;V(T^^)(l-F?/+  yV(l  -ar^)(l  -A'-a^) 
l-Jc'x'y^ 

r rf^ I dy_ 1 

+  2A;2.<-^  [V(l-r)(l-A;-'r'y-  ^^ 


+  V(l  - iK=^)(l  -  A;^x-^) .  (/^]  I  =  C. 
Keduciuy;,  by  the  aid  of  (2)  and  (3),  we  have 

W(l-/)(l-A•^v'')  +  ?/V(l-ir')(l-F^^^ 


Chap.  XVI.]  ELLIPTIC    INTKCiRALS.  245 

To  determine  C,  from  (4)  y  =  C  when  a- =  0,  and  from  (1) 
y  =  c  when  x  =  0.     Therefore  C  =  c,  and  we  get 

F{k,x)-\-F{X-,ij) 

~     V'"  1  -  k^x^y'  /         '-••J 

our  required  addition  formula. 

An  addition  formula  for  E  (fc,  x)  can  be  obtained  in  very 
much  the  same  way,  but  the  work  is  rather  complicated,  and 
it  is  better  to  use  a  method  which  will  be  explained  later. 


THE    ELLIPTIC    FUXCTIONS. 

191.  We  have  just  seen  that  there  is  an  analogy  between 
the  Elliptic  Integral  F( A',  .t)  ,  and  the  familiar  functions  logx, 
sin"^a;,  and  tan^x;  and  we  know  that  the  theory  of  these 
functions  is  ultimately  connected  with  that  of  their  inverse 
functions,  log~^rt  or  e",  sin?<,  and  tanu;  and,  indeed,  that  the 
latter  are  so  much  sim[)ler  than  the  former  that  it  is  customary 
to  regard  them  as  the  direct  functions,  and  the  logarithm,  tiie 
anti-sine,  and  the  anti-tangent  as  the  inverse  functions. 

For  example  :  the  first  three  addition  formulas  just  obtained 
are  much  simpler  when  we  express  them  in  terms  of  the  direct 
functions,  and  they  become 

log  \il.  +  I')  =  log  ' u  •  log  'u, 

or  gC'-H-")  —  e" .  e»^  [Ij 

sin(»/  +  v)    =  sin?<  Vl  —  sin^r  +  sin  v  Vl  —  sin^M  ; 

or  sin('<-|-f)    =  sin  »<  cost*  +  cos  ?<  sin  y,  [2] 

.      /      ,     V  tan?<4-tanv  ran 

tan(»<  +  r)   =  ^      ;  [3] 

1  —  tan  H  •  tan  o 

and  in   this   form   they   seem   to   better  deserve  the   name    of 
addition  formulas. 


246  INTEGRAL  CALCULUS.  [Art.  192. 

In  the  same  way  the  addition  formula  for  F{k,  x)  can  be 
more  simpl}'  written  in  terms  of  the  function  which  we  might 
naturally  represent  by  F  he  (mod.  k)  ;  and,  as  we  might 
expect,  this  function  has  many  interesting  and  important  prop- 
erties which  well  deserve  investigation. 

Since  in  most  of  the  work  which  follows  we  shall  generally 
employ  the  same  modulus  throughout,  we  shall  not  take  the 
trouble  to  write  it  except  in  the  few  cases  where  its  omission 
might  give  rise  to  confusion,  and  then  we  shall  put  (mod.  k) 
after  the  function,  as  above  with  F'^u  (mod.  k),  or  we  shall 
write  it  more  briefly  as  F^'^(u,  k). 

192.  In  Arts.  ITS  and  179  we  have  adopted  two  forms  of 
notation  for  an  Elliptic  Integral  of  the  first  class,  F{k,  x)  and 
F{k,  ct>)  ; 

F(k,  x)=  r        ^^^ 

.A.  V(l  -x-){l-k''x^) 

F(k.^)=r    ''^   -=r^. 

Jo   ^1  —k'^sm^^     -'"    A</) 

where  x  =  sin  <^,  Vl  —  x-  =  cos  <^, 

and  Vl-A;2a^  =  Vl-^-sin^*^  =  A<^. 

If  we  let  u  =  F {k,  x)  =  F(k,  <f>) , 

we  have  in  Art.  179  called  ff>  the  amplitude  of  n,  and  sin0, 
cos<^,  and  A<^  may  be  called  the  sine,  the  cosine,  and  the  delta  of 
the  amplitude  of  u  ;  and  (/>,  sin^,  cos<^,  and  A^  may  be  written 
amw,  sinam»,  cosamvt,  and  Aamw,  or,  more  briefly,  sxmu,  sn?/, 
cn?<,  and  dun  ;  and  may  be  read  amplitude?*,  sine  amplitude ?«, 
cosine  amplitudes,  and  delta  amplitude  it.  Formulating,  we 
have 

u  =  F{k,x)  =  F{k,<l>),  ^ 

<!>  =  am?t, 

a;  =  sin  ^  =  sn  ?t,  )■  ^  [1] 

Vl  —x^=  cos  <^  =  en  ?<, 
Vl-Ar^a;'=A<^  =  dnw, 


Cll.vr.  XVI.]  ELLirXrC    INTEflKALS.  247 

suK,  cnu,  duM,  are  trigouotuetric  fuuclious  of  <^,  the  ampli- 
tude of  w,  but  the}'  may  be  regarded  as  new  aud  somewhat 
complicated  fuuctious  of  u  itself,  aud  from  this  point  of  view 
they  are  called  Elliptic  Functions  of  u. 

amn  also  is  sometimes  called  an  Elliptic  Function  ;  and  there 
are  various  allied  functions  that  are  sonK-times  included  under 
the  general  title  of  Elliptic  Fuuctious.  We  shall,  however, 
restrict  the  name  to  sn  »,  cu ;/,  and  dn  «.  They  have  an  analogy 
with  trigonometric  functions,  and  have  a  theory  which  closely 
resembles  that  of  trigonometric  functions,  and  which  we  shall 
proceed  to  develop.  It  must,  however,  be  kept  in  mind  that 
the  independent  variable  n  is  not  an  angle,  as  in  the  case 
of  the  trigonometric  fuuctious. 

Of  course,  with  our  notation,  u^F{J:,  .r)  =: sn~ ' (./;,  k),  or 
u  =  F  (k,  <f>)  =  am-  •  (<^,  k). 

The  fundamental  formulas  connecting  the  Elliptic  Functions 
of  a  single  quantity  follow  iunuediatcly  from  the  defiuilious 
[1],  and  are 

su'??  +  cn-(/  =  1,  [2] 

dn-'»  +  /t-su^<  =  l,  [3] 

[4] 


cl  am  u 
da 

= 

du», 

clsnu 
dn 

= 

en '/  .  ( 

n  u 

d  en  11 

^ 

—  sn  n 

.di 

1  u. 

da 
rfdn« 


[5] 
[6] 

=  — A"sn«.cuM,  [7] 


da 
The  only  one  of  this  set  which  needs  any  explanation  is  [H 


248 
hence 

and,  finally, 

Since 
we  see  that 


INTEGRAL   CALCULUS. 


[Art.  103, 


A0  dnu 


d  am  II 
(In 


—  (ln». 


Jo      A<^      Jo    A(— ^)  Jo   A^' 


am(  —  ?/)  =  —  am?(, 
sn  {—  ii)  =  —  sn  », 
en  (—  u)  =cnn, 
(In  (—  ti)  =dnt;, 


[8] 


That  is,  sn?<  is  an  odd  function  of  u,  and  enw  and  dnu  are 
even  functions  of  w. 


Since 
we  have 


X 


A<^ 


0, 


am(0)  =  0, 
sn(0)  =0, 
cn(0)  =  1, 
dn(())  =  1,  J 


[9] 


193.  Our  addition  fonnula  for  the  sine  amplitude  flows 
immediately  from  [;">],  Art.  190.  Let  u=  F{k,x)  and 
v  =  F{k,y),  and  take  the  sine  amplitude  of  each  member  of 
[5],  Art.  190;  we  get 


sn  (n  -\-v)  = 


snn  .  cn^  .  dn^  +  cnu  .  snv  .  dnu 
1  —  /i^  .  sn-«  .  sn^v 


If  now  we  replace  v  by  —  r,  and  simplify  by  [8],  Art.  192, 
wc  have 

,           s       snw  .  en?' .  dn?;  —  cnw  .  snu  .  duM 
sn('f  — r)  =       — — , 

and  the  two  formulas  can  be  com])ined  if  we  use  the  sitrn  ±  ; 


Chap   XVI.]  ELLIPTIC    IXTKGRALS.  249 

y     ,    ,^  _  sn u  .  cu i^ .  du V  ±  en ic.snv .  dnu  p., 

1  —Jir.  sn-(( .  sir-u 


From  [1],  with  the  aid  of  [2]  and  [;3],  Art.  192,  wo  can  get, 
after  a  rather  ehiborate  reducliou,  tiie  aildition  foiuuilas  for 
en  and  du. 

/    j_    \       enw  .  cnv  ^F  sn?( .  sn-y  .  diiM  .  dnv  p,, 

1  —k^ .  sn-»  .  sn-y  *•  -^ 

,    ,      .     .       dn?<  .  dur  T  i"^ .  sn»<  .  snv  .  cn«  .  cnu  tot 

dn  (u  ±  I')  = f^ ,^ _ [3] 


From  [1].  [2],  and  [3J  a  huge  iiumlier  of  formuhis  can  he 

readily   obtained.      AVe   give    only    those    for   sn ;    there  are 
similar  ones  for  en  and  dn. 

....        .          x                   2  sn  It .  en  V .  dn  V  r-.-, 

sn(?<  +  t')  +  sn(«- y)             =  -, ^ -•  [4J 

1  —  ^- .  sn- ?f  .  sn^y 

,     ,     X           /          X                  2cnw.snv.dnw  r-i 

sn  (u  +  v)  —  sn  (u  —  v)             = — — •  [ol 

^           ^           ^           ^                 1  -A;-.sir«.8n2y  ■■  "^ 


sn(M  +  v).sn(w-'y)  =- — •  [0] 

1  —  AT  •  sn-  II  •  sn'*  /• 

,    ,        /     ,     \         /  \  cn^v  +  sn-u .  du'y  r-n 

1  —  A"' .  sn'^M  .  sn'^v 

1  +  k- sn  ( H  +  V)  .  sn  ( u  -  r)     =  — — T:; j -. [8] 

1  —  A-  .  sn*?/  .  sn^y 


n    I       /      I     \n  n   1       /  \-i       (cn  r  +  sn '/ .  dn  c)-'  ^oi 


From  [2]  and  [.'5]  conies  tlic  uscfnl  fonniihi 

cu(»  +  v)  =  cnu  .  en  I'  —  sn"  .  sn  *• .  dn  ('/  -\-v).  [10] 


250                                  INTEGRAL   CALCULUS.  [Art.  194. 

194.    If  in  formulas  [1],  [2J,  and  [3]  of  Art.  193  we   let 

v  =  w,  we   get   the   following   formulas   for  sn  2m,   en  2m,  and 

dn2M: 

n        2sn?t .  en?< .  dn?t  p-t 

sn2u  =  — — — ; ,  [1] 

1  —  k-sn*u 


I'u  —  SIT  71 .  du^?A      1  —  2su'w  +  7c^sn*u 


1  —  A;-sn'*w  1  —  ^-sn^ 


[2] 


,    n        dn^M  —  A;- .  sn-?( .  cn-u      1  —  2k^sn^u  +  ]c^sn*tt  ro-, 

>"''" T^:^¥^u —  =  — r^¥^u —  M 

From  those  come  readily 

,  ,1         2sn-it.dn^M  r>n 

1— cn2«t  = — - — ,  [41 

l-dn2«  =  -- —— ,  [6] 

1  —  Arsn*M 

••    I    1     >  2dn^?<  PTT 

l  +  dn2u  = — — — .  [7] 


195.   Replacing  u  by     ,  and  dividing  [4]  by  [7]  and  [G] 
by  [5],  Art.  194,  we  have 

nu       1— cnit  1  —  dnw 


2       l+dn?t      A;^(l+cnM) 


[1] 


jU  _dnM+  en»  _  —  Jc' ^  +  Jc^ en u -{- dmi  r-oT 

^^2"    l  +  duM    ~         A-(l+cutO        ''  *-  -^ 

,  2U  _  k'^  +  dnM  +  k'^cnu  _  (en  u  4-  dn  u)  pg-i 

'^  2~  1  +dn?t  ~     (1  +vnii)    '    '  ^  ^ 

where  A'^=l— ^'j  and    is   the  square  of   the  complementary 
modulus. 


r  iiAP.  XVI.]  ELLIPTIC    IXTECRALS.  2ol 

From  [1],  [2],  and  [13],  we  can  get  without  diniculty  the  set 

o»  dn?<  — enu  p-t 

2      k'-  +  dmc  —  k-cnu 

2       ^'-'  +  dn«-fc"^cuw  •-  -■ 

^^2Tf^_^^^(l+dn«)_,  j-g-j 

2      k'^  +  dmi  —  k-cnu 

Numerous  additional  formuhis  can  be  obtained  by  the  exer- 
cise of  a  little  ingenuity,  but  we  have  given  the  most  useful  and 
important  ones,  and  they  form  a  set  as  complete  as  the  usual 
collections  of  trigonometric  formulas. 


Periodkiti/  of  the  Elliptic  Functions. 

196.    We  have  seen  (Art.  18G,  [2])  that 

F{k,n7r+p)  =  2nK+F{k,p),  [1] 

where  A"  is  the  complete  Elliptic  Integral  of  the  first  class. 

Let  ?<  =  F{k,  p),  and  take  the  amplitude  of  each  member  of 
[1]  ;  we  get 

am  ( ?/  +  2  )t  K )  =  n  tt  +  am  (/ ;  [2] 

or,  replacing  n  by  2?j, 

am  ( "  +  1  n  K)  =  2  ?j  TT  +  am  u  ;  [3] 

whence 

sn  {it  4-1  /(  A')  =  sn  »/,  ^ 

en  («  4--i»  A')  =  cu's   I;  [4] 

dn  (?t  +  4?j  A')  =  du'/,  j 

and  sn  */,  cii  »/,  dii  u  are  periodic  fimctions.  and  have  the  real 
period  1  A',  dn '/  actually  has  tlu'  smaller  period  2  A'.  :is  mny 
be  seen  Ity  taking  the  delta  of  both  members  of  [2] 


252 


LNTEGKAL   CALCULUS. 


[Art.  li.^. 


Since  the  amplitude  of  A" is  -,  we  have 


[5] 


and   our   addition    formulas   [1],  [2],  [3],   Art.  193,  give   us 
readily 


(?<  +  A)    =      -— , 
dnrt 


en  (w  +  A")    =  — 


^■'  sn  u 
dun 


du(»+Ar)    = 


dnu 


[6] 


sn(?t  +  2A')  =  - 

~snn, 

cn(?<  +  2A')=: 

-cn?A, 

dn(«  +  2A')  = 

dUK, 

sn(?<  +  .'Ur)  =  - 

cn?( 
dn  H 

cn(w  +  37r)  = 

k'  sn  V 
dn  u 

dn(u-|-;]A')  = 

dn  11 

[7] 


[«] 


sn  {u  -\-4  K)  =  sn  ?/,  ^ 
en  (71  +  1  A")  =  en  '^   l  , 
dn(?/  +  4  A')  =  dn?^,  J 


[9] 


a  confirmation  of  [4]. 


^  .1'.  XA^I.]  ELLIPTIC    INTEGRALS.  258 

197.  Jt  is  easy  to  get  formulas  for  the  sn,  cm.  :iinl  dii  of  an 
imaginary  variable,  wV— 1,  by  the  aid  of  a  titinsfonnatioii  thi'i 
to  Jacobi. 

Let  v  =  F{k,ct>)=r^^,,  (1) 

so  that  <^  =  a)nr,   sin^^sne,   and  cos<^  =  fnr.       In    (1).   re- 
place </)  by  i//,   <^  and  i//  biMn;j,-  connected  by  tlie  relation 

sine/)  =  V— i  .  tan i/',  (2) 

whence  cos^=.seci//,  (3) 


A(^  =  Vl  -  k'  sin->  =  Vl  +  A^-'tan^,  (4) 

and  (J(f)=  ^f  —  \  .  seciA  .  dip. 

Since  il/  and  4>  equal  zero  together, 

Ju   Vl-A-'-sin^'^ 
If  now  we  let  n  =  F{k',  ij/) , 
we  have  -y  =  ?t  V  —  1 .  (5) 

Hence,    since   li/ =  ain »  (mod//) ,    we    have    from    (2),    (3), 
and  (4), 

sn(r,  /.•)  =  V-1  — ) — 77(' 
^       ^  en  («,  A- ) 

en  (v,  k)  =  — ; — —i 
^       ^      en  (w,  A;') 

dn  («,  A:') 

^       ■^       en  (m,  /c  ) 

or,  as  i'  =  ?<  V—  1, 


264 


INTEGRAL   CALCULUS. 

sn  (u,  Ji') 
on  («,  k') 


sn  {u  V-  1,  A-)  =  V-  1 
en  {u  V—  1,  /.-)  = 


en  (//,  k') 


[Art.  197. 


m 


dn  (u  V—  1,  A;)  = ; jr: ' 

It   is    interesting    to  note    tliiit    if   w  i.s    replaced    in   (6)   by 
mV—  1,  the  formulas  reduce  to 

sn  (—  ii)  =  —  sn?^ 
en  (—  n)  =  cwn, 
dn  (—?;)  =  dn?<, 

and  are  still   true.     Consequently,  in   (G),  u  may  he  eitlier  a 
real  or  a  pure  imaginary. 


Let 


rfl// 


^  (mod/i') 


=./: 


d^ 


=  K'. 


Vl-A;'-sin-> 

Then,    by   Art.    19G,    4/r'    i.s    a    period    for    sn?t  (mod/i'), 
en u  (mod 7i') ,  and  dn  « (mod A') . 
Hence 

sn  (ic\/—  l  +  4?i7t'' V—  1)  =  sn((  V—  1, 

en  (u  V—  1  +  4  II  /r'  V  —  1 )  =  en  n  V—  1 , 

dn  (a  V^nr+  4  i>  K'  y/^\)  =  dnn  V^^  ; 

or,  replacing  «V—  1  by  r, 

sn  {v  +  4  /(  A''  V  —  1 )  =  sn  ?", 

en  (r  +  4  ;;  K'  V^l )  =^  en  r, 

dn  {v  +  4  ?i  7r'  V^n")  =  ^1»  ^'^ 

and  4 /iT'V—  1  is  a  period  for  sn,  en,  and  dn. 

We  see,  then,  that  our  Elliptic  Functions,  like  Trigonometric 
Functions,  have  a  real  period,  and,  like  Exponential  Functions, 
Lave  a  pure  imaginary  period.  They  are,  then,  what  may  be  called 


['] 


Chap.  XVI.] 


ELLIPTIC    INTKGKALS. 


255 


Doubly  Periodic  Functions,  and  they  are  often  studied  from  the 
point  of  view  of  their  double  periodicity. 

Like  Trigonometric  Functions,  the  E^Uiptic  Functions  may  be 
developed  in  series,  and  from  these  series  their  values  may  be 
computed,  and  tables  resembling  Trigonometric  tal)les  may  be 
prepared. 

A  partial  three-place  table  is  here  [)resented  as  a  sample.     It 

is  complete  for  Elliptic  Functions  having  the  modulus  ~  ;  that 

is,  0.7. 

Modulus  —  =  0.7. 
2 


n 

snu 

cni< 

dnu 

0.00 

0.000 

1.000 

1.000 

0.05 

0.051 

0.9</; 

0.999 

0.15 

0.150 

0.989 

0.994 

0.25 

0.247 

0.969 

0.985 

0.35 

0.340 

0.940 

0.971 

0.45 

0.429 

0.903 

0.953 

0.55 

0.512 

0.S.S9 

0.932 

0.65 

0.5S9 

0.808 

0.909 

0.75 

0.659 

0.752 

0.885 

0.85 

0.722 

0.692 

0.860 

0.95 

0.77S 

0.628 

0.835 

1.05 

0.827 

0.562 

0.811 

1.15 

0.869 

0.494 

0.789 

1.25 

0.906 

0.424 

0.768 

1.35 

0.935 

0.353 

0.750 

1.45 

0.959 

0.284 

0.735 

1.55 

0.977 

0.213 

0.723 

1.65 

0.990 

0.143 

0.714 

1.75 

0.997 

0.072 

0.70«; 

K.  1.85 

1.000 

0.000 

0.707 

From  this  table,  by  the  aid  of  formulas  [4],  [G],  [7],  and 
(8)  of  Art.  rjG,  sn»,  en?<,  and  dnu  may  be  reailily  obtained 

for  any  value  of  «  if  the  modulus  is  -^- 


256  INTEGRAL  CALCULUS.  [Art.  198, 

As  a  matter  of  fact  no  complete  set  of  tables  for  the  Elliptic 
Functions  lias  been  published,  and  their  values  are  usually  ob- 
tained indirectly  from  Legeudre's  Tables  of  Elliptic  Integrals 
(v.  Arts.  186,  187),  unless  especial  accuracy  is  required,  in 
which  case  they  must  be  computed  by  methods  which  we  have 
not  space  to  give. 

198.  The  Elliptic  Integral  of  the  second  class  J5^  (A-,  <^)  can 
be  expressed  in  terms  of  f^Uiptic  Functions,  and  for  some 
purposes  there  is  a  decided  advantage  in  the  new  form. 


X<i> 


We  have  E{k,  4>)=  i    A<^  .  defy. 

Let  u  =  F(k,  (f)) ,  then  eft  =  am  u,  and  E  (A-,  <^)  may  be  written 
E{k,  amu),  or,  more  simply,  E{amu),  if  the  modulus  can  be 
omitted  without  danger  of  confusion. 

Xani  II 
du({ .  d avail ; 

or,  since  by  (4),  Art.  l'J2, 

damn  =  du^;  .  du, 


E 


(am?t)=  I    dn-)'.f?u.  [1] 

As  an  example  of  the  usefulness  of  the  form  just  given  in 
[1],  we  will  employ  it  in  getting  an  addition  formula  for 
Elliptic  Integrals  of  the  second  class. 

E{&mu)  +  E  (i^mv) 

=  I    dn-n.<?'f+  I    dii'v'.fZu 

=  I    dn'Z  .  (h  +  I    i\n-z.dz 

dn'-z  .  dz  4-1    dn-2; .  dz  —  I    dn-z  .  dz 

=  £'[am  (n  +  (•)] -f  I    ihi'z.dz—  |  dn'z.dz. 


Chap.  XVI.]  ELLII'TIC    INTKCII.M.S.  2/>7 

l\ei)laciiig  z  by  u  +z,  and  iviueinln'ring  that  u  and  v  are  given 
constants, 

I    dn'-2  .dz=  I    dn-  {u-\-z)  dz, 
and 
E{amii)+  E(iiu\v)  = 

E  [am  (((  +  f)]  -  pr.'ln-  ( n  +  z)  -dn-^]  dz.         (2) 

dn-(M  +  2;)— dn-2;=  [dn  {n  -fs)  +  dn^]  [dn  (;<  +2;)  — duz].   (:i) 

We  can  obtain  from  [3],  Art.   lt)o,  the  foUowing  formuhi.s 
analogous  to  [4J  and  [5],  Art.  193, 

1    /      ,     \  ,    1    /  \  2  dn  ?( .  dn  v  ... 

dnO<  +  iO  +  tlii(H-'y)=^ -^^; —,  (4 

1  —  Arsu-u  .  sn-v 

,    ,      ,     .       1    .  X  2A,-^sn?t .  snv.  cnw .  cnv 

1  —  A-sn-»( .  sn-v 

If  in  (4)   and  (.">)   we  let  ?t  +  v=(t+2;,  and  u  —  v  =  z,  and 
substitute  tiie  results  in  (3),  we  get 

dn^  {ii  +  z)  —  (\\YZ 

4  A-- su  (  -  +  2  ) en  (  ^  +  ^   (In   -  +  2;  J sn  _^  en  ^  dn  - 

1  —k-su-'^sn-r^  +  zj 
and 
I  [dn-  (u  +z)  —  dn-z]  f/2 

=  — 2sn    c-n    dn      I 7= ^ ■: r^. 

„      u      u  ,   u 
2sn    on    dn- 

2       2       2  1 


su'--  1  —  Jc- an- 

2 


U       .,(\L    ,       \ 


258  INTEGUAL   CALCULUS.  [Art.  1'.)9. 

since  -  2  Jc"  sir"  sn  /^"  +  A  <-n  f''^  +  z\  dii  /"'*  +  A (?2  is  the  differ- 
enti 


1  of  1  —  A-  sir-  su- (  -  +  2  V 
2        V-'        J 


-  CM  -  au  - 

•'    ''    H ! I "1 

■-  1  —  /.-  sir  _  sir  7+1^         1  —  A-  sn*  - 


sn 


A-2sn-cn-dn-         su-       +y    —  sn''- 
2       2       2  V2         /  2 


1  _  A--  sn*  -  1  -  A--  sn^ "  su^  f  "  +  v 

2  2        V2 

=  —  k- .  sn  11 .  sn  r  .  sn  (?<  +  i')  ■, 

by  (i),  Art.  194,  and  [(>],  Art.  193. 
Hence  by  (2), 

E  (nmu)  +  £'(ami')  =  E[am(ii-{-  r)]  +  ^"snit .  snv  .  sn(if  +  t'), 
our  required  addition  foruuihi. 


APPLICATIONS. 

Rectification  of  the  Lemniscate. 

199.  From  the  polar  equation  of  the  Lemniscate,  7-^=  a^  cos  2^, 
referred  to  its  centre  as  orijj;in  and  its  axis  as  axis,  we  get  as 
the  length  of  tiie  are,  measured  from  the  vertex  to  any  point, 
P,  whose  coordinates  are  r  and  6. 

s  =  a  \    ==  a  I  — .  [1 J 

-^"    Vcos2^        -^^  Vl-2siir^ 


CiiAP.  XVI.J  KiJJi'Tic   inti:(;i;ai.s.  2r)0 

and   foi-  the  arc*  of  tlu'  quadrant  of  the  Li-nmiscate,  that  is,  the 
aic  from  vi-rti'X  to  conliL', 


^^-  c-^] 


^"  VI 


These  diflfer  from  Klli[)tie  Integrals  of  the  first  chiss  only  in 
that  the  coefficient  of  sin-^  is  greater  than  nnity,  and  tliov  may 
be  reduced  to  the  standard  form  by  a  simple  device. 

Introduce  in  [1]  <^  in  place  of  6,  <(>  and  6  being  connected  by 
the  relation  sin-d)  =  2  s'urO. 


I'lieu  we  have  Vl-  2  sin-/^  =  cos<^, 

J2_ 


and  ^^^^V2      cos<^fZ<^ 


Hence      s  =^^  C -^^— ^^^f^^.A 

•2   ^'   Vl-^sin^'ci         -2        V  2    V 


[••'] 


ayJ-2  C'__d±__  ^  aV2 ^AV2  7r\  *    |- , j 

2   -''^  ^/^^±'sin^^         -2        V  2  '  2/ 


and  ..  .        

Vl  —  4  sin-0 


The  auxiliary  angle  cfi  is  very  easily  constructed  when  the 
point  P  of  the  Lemniscate  is  given.  We  have  r  =  aVcos2^, 
and  we  have  seen  that  Vcos  '26  =  cos</) ;  hence  r  =  a  cosf/b.  If, 
then,  on  a  as  a  diameter  we  describe 
a  semi-circumference,  and  with  the 
centre  0  of  the  Lemniscate  as  a 
centre,  and  with  a  radius  equal  to  r, 
we  describe  an  arc,  and  join  with  0  qL 
the  point  Q  where  this  arc  intersects 
the  semi-circumference,  the  angle  ma«le  Ity  OQjw'ah  n  is  equal 
to  (f>.     For  OQ  =  a  cos  AOQ  and  OP  =  a  Vcos  2 6. 


260  INTEGRAL   CALCULUS.  [Art.  !<.»;). 

EXAMI'LKS. 


(1)    Find  the  nunierieiil  value  of  i 


(2)    Kediicc   ('  ^^"^  to   nn   Elliptic  Integral  of    tl 


Jo  Vl-4sin-<i 

Ans.  0.843. 
^•*  d^ 

h   Vl  — wsin'^^ 
first  flass,  AvluMi  v  >  1. 

jitis    J—    (    ^  whore  sin'  (//=  n  &\\\-  <^. 

(3)  Tlie  half-axis  of  a  Lemniscate  is  2.  What  is  the  length 
of  the  aic  of  a  quadrant?  of  the  arc  from  the  vertex  to  the 
point  whose  polar  angle  is  30°?  Ans.  2.622  ;   1.168. 

In  the  inverse  problem  of  cutting  off  an  arc  of  given 
length  the  Elliptic  Functions  are  of  service.  As  an  interesting 
example,  let  us  find  the  point  which  bisects  the  quadrantal  arc 
of  the  Lemniscate. 


<fi)- 


Here  s  =—-—  \F' 

and  we  wish  to  find  c^  and  then  6 

Let  '^  =  ^(f'^' 

■"■  1  t\  ^     ^  V  2 

am?/  =    ,  sn  ii  z=  \.  cn?<  =  0,  and  tin  n  =  — • 
2  2 

IJy  [1]  and  [2],  Art.  l!»r», 

oW       1  — cn?t  oU      dnw  +  cnw 

sn^-  = ,      en-- = • 

2      l+duM  2         l+dn?i 


Therefore, 


B\r  - 

2_  g?(_     1  — cn?<     _    '    _    /- 

2^  2      dnu-}-cnn      V2 

2  ~T 


Chap.  XVI.] 


ELLIPTIC   INTKGllALS. 


201 


1[\  thou,  the  reciuiied  amplitude  is  (f>, 
tiui-</)  =  V2, 


and 


tan<^ 


Siuce  sin^<^  =  2  sin-^,  we  cau  compute  0  without  difficulty, 
and  so  get  our  required  point.  If,  however,  a  construction  will 
suffice,  a  very  simple  one  gives  the  point. 

Erect  at  A  a  perpendicular 
whose  length  is  a  mean  pro- 
portional between  a  and  a^/^2. 
The  angle  subtended  at  0  by 
this  perpendicular  is  (f>.  and 
the  corresponding  point,  P,  is 
found  by  the  method  described  0 
on  page  255. 


Rectification  of  the  Ellipse. 

200.    We  have  seen  in  Art.    177  that  the  length  of  an  arc 
of  an  Ellipse  measured  from  the  end  of  the  minor  axis  is 


■'x-2 


dz. 


[1] 


If  we  let  X  =  aaluff),  [1]  becomes 

s  =  a  (''' V 1  ^^siu-</>  .  di>  =  a E (e,  ^ ),  [2] 

e,  the  modulus  of  the  Elliptic  Integral,  being  the  eccentricity 
of  the  Ellipse.  If  x  =  a,  <^  = '^,  and  the  length  of  the  KUiptic 
quadrant  is  n 

s^  =  (,  ("V 1  -^'='sin'^  .  <l<i>  =  a  E  (e,  -\  [3] 


2(32 


INTEGRAL    CALCULUS. 


[Art.  201. 


The  length  of  an  arc  of  the  Elliptic  quadrant,  not  measured 
from  the  extremity  of  the  minor  axis,  can  of  course  be  ex- 
pressed as  the  difference  between  two  Elliptic  Integrals  of  the 
second  class. 

The  amplitude  <;^,  corres|)onding  to  a  given  point  P,  of  the 
Ellipse,  is  easily  constructed  as  follows  :  On  the  major  axis 
as  diameter  describe  a  circumference ; 
extend  the  ordinate  of  P  until  it  meets 
the  circumfei'ence,  and  join  the  point  of 
intersection  with  the  centre  of  the  ellipse. 
The  angle  the  joining  line  makes  with 
the  minor  axis  is  seen  to  be  the  required 
amplitude  <^.  If  <f>  is  given,  P  may  be 
found  by  reversing  the  order  of  the  steps 
of  the  construction. 


EIXAMI'I.KS. 

r-       ?/-  '" 

The  equation  of  an  ellipse  is  ^ — |- i-  =  1    required  the  length 

of  the  quadrantal  arc  ;  of  the  arc  whose  extremities  liave  the 
abscissas  2  and  2V2.  ^l)(.s.  ,").4  ;  O.ltl  1. 


(2)   Find   the  abscissa  of  tiie  end  of  the-  unit  arc  measnri-d 

from  tiie  extremity  of  the  minor  axis  in  the  ellipse  ^—  +  ---  =  1  ; 

of  the  point  which  bisects  the  arc  of  the  quadrant. 

Ans.  O.OyC;  2.57. 

201.    \\y  the  aid  of  the  addition  formula 

E{-MnH)  +  E{i\\nv)=E  [am  (n+  w)]  +7i;-snusu  v&n{n+v) 

([G],  Art.  198) 


it  is  always  possible  to  find  an  arc  of  an  ellipse  differing  from 
the  sum  of  two  given  arcs  by  an  exi)ression  which  is  algebraic 
in  terms  of  the  abscissas  of  the  extremities  of  the  three  arcs. 
Tliis  will  bi!  clearer  if  we  modify  slightly  the  form  of  our  addi- 
tion formula. 


Ciivr.  XVr.]  KI.LIl'TK"    INTKfMtALS.  263 

Let  c/)  =  :im»,       i// =  tun /",       and  fr  =  aiii('/ +  r). 

Thou  tho  forimila  given  above  becomes 

E{h\  cfi)  +  E{k;  4,)  =  E{h\  o-)  +  A-- sine/)  sin  i/zsino-,        [1] 

where  ^,  ip,  and  <t  are  three  angles  connected  Jty  the  rehition 

coso-  =  cos<^  cost/'  —  sin  eft  sini/'Arr.  [2] 

by  [10].  Art.  l'J3. 

If  we  multiply  [1]  by  a  aud  take  A-  e(iual  to  e,  we  get 
aE  ((%  <^)  +  aE  {(>,  ^)  =  aE  (c,  a)  +  —;x\  .  x., .  x^, 

if  a'l,  iCo,  and  x.^  are  the  abscissas  of  the  points  whose  amplitudes 
are  (f>,  \f/,  and  a. 

The    mos^    interesting   case    is    when    it  =    ,    in    which    case 

aE{e,  cr)  is  the  arc  of  a  cpiadiant.      [2]  tlien  reduces  to 

0  =  cos<^  cost//  —  sin<^  siui/^  V 1  — e^, 
or  -  siu</)  siui// =  cos(/)  cosi/', 


tau<^  taut/' 


[3] 


and  we  get  from  [1] 

aE{e^  (i>)—\aE  (e^'^\—aE{e,  \p)    =  (f*'- sin<^  siin/'.        [1] 

If,  then,  any  point,  7*,  is  given,  [.'^]  will  enable  us  to  get 
the  amplitude  of  a  second  point,  Q,  and 
thus  to  find  Q,  Q  and  P  being  so  re- 
lated that  the  arc  Bl\  minus  tl»e  arc  AQ, 
shall  be  equal  to  a  (piantity  which  is 
proportional  to  the  product  of  the  ab- 
scissas of   /'  and   Q. 


264  INTEGRAL   CALCULUS.  [Art.  202. 

For  the  special  case  wliero  ^  and  ip  arc  equal  we  have  from 


V^l 

tan  ^  = 

la 

and  from 

m 

BP 

-AP  = 

■■  (le-  sin 

=  0  = 

a- 

-b 

This  point,  which  divides  the  quadrant  into  two  arcs  whose 
difference  is  equal  to  the  difference  between  the  semi-axes,  has 
a  number  of  curious  properties,  and  is  known  as  Fagyiani's 
point. 

Examples. 

(1)   Show  that  the  distance  of  the  normal  at  F^agnani's  point, 
\         from  the  centre  of  the  ellipse,  is  equal  to  a  —  h. 

(^     y       (2)   Show  that  the  angle  between  the  normals  at  P  and  Q,  in 
.     N     the  figure  is  equal  to  i//  —  ^  ;  tiiat  tlie  normals  are  equidistant 
*^      from  0 ;  that  this  distance  is  BP  —  AQ. 

Rectification  of  the  Hyperbola. 

202.  If  the  arc  of  the  Hyperbola  is  measured  from  the 
vertex  to  any  given  point,  P,  whose  coordinates  are  x  and  y, 
its  length  is  easily  found  to  be 

■\dy,  [1] 


b- 
or 


=s:-7r>  ^^^ 


•  +i.  / 


if  e  is  the  eccentricity  of  tiie  Hyperbola.     Let 

ae 
b- 


ae         .       , 


Chap.  XVI.]  ELLIPTIC   INTEGRALS.  265 

and  [2]  becomes 

S  =  I       ^=  ; 

aeJo      I         1    .  2  J 
^1  --sm^<j> 

hence  .  =  ^  r^^^^A^±^  =  !l  T  «M^!!i^  r-n 

aeJo  Vl-A;-sin2</)      ^'^Jo        A^^ 

if  ^•=-- 

e 

X  1    _  1  —  ^''       1   _      1  1  —  Ar'  siu-  <t>  —  Jr  cos-  <^ 

A</)  ~  1  -  A--  '  A(^  ~  1  -  A'-  '  A(/) 

1      ^Ac^-^'^'"'^'' 


1  -  ]r\  A(/) 


and  .=  ^  .  -^-f  f  sec^<^A<^r/<A-^^  f^l 

ae     1  —  A-|_il/o  c/o   A<^J 

=-  •  -^;,r  rsec2<^Ac/.f/<^-/.--'F(/.-,  <^)1 
ae     1  —  A-|jl/o  J 


If  we  integrate  l»y  parts, 

J"  sec2<^  Ac^  <lcf>  =  tan  </)  A<^  +  /.-'  j     ^-^-^(l<t> ; 
0  ./o      A</> 


but  A-^^iirj^  =  -L_A<^ 


A</) 

-A<^, 
A<^         A<^ 


and  A:^  r*sin^  _  ^^^^  ^  ^.,  ^ ,.   ^^^  ^  _  ^;  ^  /.^  ^^  ^ 

c/o       A<^ 

Hence 

.s-  =  -^(A-,  ,h)  -     -^'    ,,..  [/;  ( A-,  </,)  -  tan  ^  A</,J. 
rte  «f  (1  —  A-) 

But  1  -  Ar^=  4^,. 


llSTEdRAI.    CALCUI.US. 


[Art.  202 


therefore 


F{1\  ^)-aeE{k,  (p)  + (letancfy^cf,, 


s  =  ^f(-,  ^  -  (U'E  (-,  <p\  +  ae  tan </>  A<^.        [4] 
The  angle  ^  corresponding  to  a  given  point  P  is  easily  con- 


structed.    AVe  have  only  to  erect  a  perpendicular  to  the  trans- 
verse  axis  at  a  distance  —  = 


from  the  origin  ;  that  is, 

ae      Vet-  +  b- 

at  a  distance  from  the  centre  ecpial  to  the  projection  of  b  on 
the  asymptote,  and  to  join  the  projection  of  P  on  this  line  with 
the  centre.  The  angle  made  by  the  joining  line  with  the  trans- 
verse axis  is  <fi,  for  its  tangent  is  clearly 


b- 


y/a'  +  V 


Examples. 


(1)   Find  the  length  of  the  arc  of  the  hyperbola 


IG       9 
measured  from  the  vertex  to  the  point  whose  ordinate  is  2. 

Ans.  2.194. 

(2)  Show  that  a«  tan  </>  A<^  is  the  distance  from  the  centre  to 
the  normal  at  P. 

(3)  Show  that  the  limiting  value  approached  by  the  difiference 
between  the  arc  and  the  portion  of  the  asymptote  cut  off  by  a 
perpendicular  upon  it  from  P,  as  P  recedes  indefinitely  from 


Cuw.  XVI.] 


ELLITTir    INTKGUALS. 


2i\l 


ferred  to  as  the  (Ufforenee  between  the  length  of  tiio  infinite  :irc 
of  the  hyperbohi  and  the  length  of  the  asymptote. 

Show  tliat  in  example  (1  i  this  difference  is  ecpial  to  2.803. 


The  Poirlubim. 

203.  We  have  seen  in  Art.  17G  that  if  a  pcndulnm  starts 
from  rest  at  a  point  of  its  arc  whose  distance  above  tlie  lowest 
point  is  i/o,  the  time  reqnired  in  rising  from  the  lowest  point  to 
a  point  whose  distance  above  the  lowest  point  is  //,  is 


9^'"    Vl-^•'sin-> 


where         k  = 


and  sin  d>  = 


\|- — .  mill    siii<p^-v| — • 

In  the  figure  let  ^1    be  the  lowest  point  of  the  arc-,   7>  the 


highest  point  reached  l»y  tlie  pt-ndiilinn,  and  /'  tiie  point 
reached  at  the  expiration  of  tlic  time  t.  Call  A(JB  a,  and 
AOP  0. 

Then    ^  =  1  -  cos«.   and   J-'^  =  VA  ( I  -cos«)  =  sii>"  =  A". 
(I  \2a  "  2 

Consequently   the    modulus   of   the    Kllii)tic   Integral  in   [1]   is 


268 


INTEGRAL   CALCULUS. 


[Art.  203. 


the  sine  of  one-fourth  the  angle  tlirougli  which  tlie  penduhim 
swings. 

y=  1  -cos^, 
a 


and 


and 


^2  a 


Vi(i 

—  cos^)  =  sin  - 

it 

Y     .  e 

\  2  «             2 

1  '/(I           •     a 
\2a             2 

and  therefore  the  sine  of  the  amplitude  of  the  Elliptic  Integral 
iu  [1]  is  easily  computed  when  the  angle  through  which  the 
pendulum    has   risen    is   given.      When    6  =  a,   sin^=l,   and 


C/.: 


;  so  that  the  time  of  a  half-oscillation  is 


i^i^it)' 


a  confirmation  of  [7],  Art.   17(!.     The  construction   indicated 


in  the  figui'c  gives  the  angle  <fi,  corresi)onding  to  any  given  arc 
A  P.     For 


JL=i-c'osAO'Q, 


and 

Therefore  ACQ  =  </> 


.    AO'Q 

:     ^T^      BUI    J_ i       • 


(1  -cos^0'g)  =  sin^^^  =  sin.4CQ. 


Chap.  XVI.]  ELLIPTIC   INTE(JUALS.  2n9 

It  is  voiy  easy  to  express  the  angle  0  in  terms  of  t. 

1(1  ,-,/  .     n 


We  have  '  ~  V    -^'^1  ^^'^^  v»  <A 


heuce  *\|        -^(  si"",  <^ 

i 

in  -  =  sin'^sn  [  ?\r- )(  mod  sin- j ; 

OS-  =  (hi(  <\^  )[  modsin"  ), 
2  V   \aj\  2/ 


sin<^  =  sn[  ^'  '^ 


and 


then 


and  sin^  =  2  sin"  snf /\|^  Jdn  [^\/-' ](  mod  sin- )• 

2      V   ^V       \  ^«/V  V 


EXAMIM.KS. 

(1)  A  penduUim  swings  through  an  angle  of  lfiO°;  required 

the  time  of  oscillation.  ^         ,  „         /a 

^I/(.s.  ;3.7()8x-- 
^^ 

(2)  Compare  the  times  required  by  the  pendulum  in  Kx.  (1) 
to  descend  through  the  first  30°,  the  second  30°,  and  the  third 
30°  of  its  arc  respectively. 

Ans.   1.028  vK  0.146.1-;  0.380  J-- 
\y  \,j  \g 

(3)  The  time  of  vibration  of  a  penduhim  swinging  in  an  arc 
of  72°  is  observed  to  be  2  seconds  ;  how  long  does  it  take  il  to 
fall  through  an  arc  of  0°,  beginning  at  a  point  2n°  from  the 
highest  point  of  the  arc  of  swing?  Ans.  O.O'Jo  seconds. 


-.KT^t^^^-?) 


r<; 


270  INTKCJKAL   CALCULUS.  [Akt.  203. 

(4)    A  peiiduluii)   for  wliich  \      has  l)een  (letcrmiiu'tl,  and  is 

equal  to  ^,  vibrates  tliioiigh  an  arc  of  180°  ;  throngli  what  arc 
does  it  rise  in  tiie  first  half-second  after  it  has  passed  its  lowest 
point?  in  the  first  ^  of  a  second?  Ans.  01)°  ;  20°  G'. 

^(5)  It  has  been  shown  in  Art.  176  that  if  ?/o>2a  the 
pendulum  will  make  complete  revolutions,  and  that  the  time 
required  to  pass  from  the  lowest  point  to  any  point  whose 
distance  above  the  lowest  point  is  y,  is 

,=„  )Z  r-  ^^^ =«,g ,,.(,,»), 

\  (l!h -/o    Vl  -  A;-  sin-  </>  \  .V.'A, 

where     k  =  \\~  and  sind)=  \    •'— 

Show  that  in  this  ease  fb=    ,  and  tiiat  sin    =  sn  [  -  \  •^'"  V 
^2'  2  ^a\  2  j 


Note.  —  In  workiii<^  with  a  pendulum  it  is  often  about  as 
easy  to  compute  F  (/.-,  <i>)  by  developing  by  the  binomial 
theorem  and  integrating  two  or  three  terms,  as  to  use  a  table 
of  Elliptic  Integrals. 

f(,,,^)=r^?i — 

>  Vi  - , 


AV'e  have 


A;-sin^^ 


( 1  -  1c- sin- </) )   -=1+1- /r  sin- <^  -f  1^ A-* sin^ <^  H , 

F(A-,  <^)  =  ("''   ^_J^,„_      =  <^  -f  ^;'''  (</.  -  sin  4>  c-«)s<^) 
J»    Vl  — /i-sin->  -i 

3  I) 

—  ^ — A'sin''<^  cosc^  -f  [—A'  (</>  —  sin  (i>  cosc^)  •••• 
32  01 


CiiAP.   XVI.]  ELLIPTIC    intk(;rals.  271 


Differentiation  and  Integration. 

204.    Kewriting  formulas  [4],  [5],  [G],  and  [7],  of  Art.  192, 
wc  have 

d  am  X  =  cln  x  dx,  [  1  ] 

(/  sn  a;  =:  en  a*  dn  x  dx,  [2] 

d  Gnx  =  —  sn  X  dn  x  dx,  [3] 

d  dn  x  =  —  k^  sn  a;  en  a:  dx,  [4  ] 

we  add  o?  tn  a:  =  — 5-  (Zx.  fol 

cn'^x  "-  -■ 

By  the  usual  method  of  differentiating  an  inverse  function 
(I,  Art.  72)  we  get  readily 

d  sn-i  (x,  k)  =  --==^= ,  [6] 

d  en-^  (a-,  A-)  = ,  — ,  [ ( 1 

d  dn-i  (a-,  A-)  = -=J^==,  [8] 

dx 
d  tn- » (x,  A)  =    ,  •  [9] 

^        ^       V(l+;r=')(l  +  /.-'^a:»)  '^  "' 

[6],  [7],  [8],  and  [9]  give  at  once  a  very  valuable  set  of  formu- 
las for  integration,  namely  : 


V(l  -  x')  (1  -  /c'x') 


k'x^ 


:  en-'  (a-,  k)  =  F(k,  eos-'.r),  [11] 


n dx 

^-  ^r(i-x^)(k'^-\- 

r'-=^_  =  .ln-Vx,A-)  =  sn-'filEZ,/A 


272  INTEGRAL   CALCULUS.  [Art.  204. 

dx 


f^   , —  =  tii-^  {X,  h)  =  F 


(A-,tan-ia-).     [13] 


If  in  [10],  [11],  [12],  and  [13]  we  substitute  y  =  x^  and 
then  change  y  to  x,  we  get 

("'-=^^— ^2sn-^(V^,A-)=2i^(A-,sin-'V!^),[14] 

Jo    ^x(l-x)(l-k-'x) 

^-  ^/x(l-x)(k"  +  /rx) 

f    ,  '^"^  -  2  du-i  (V^,  /.•) 

'^^  Vx  (1  -x){x  —  k'^) 

=  2  F(^k,  sin-' ^^^^^y         [16] 

r   ,  ^"^  =  2  tn-i  ( Vx,  70 

^0    ■\/x{l+x){l  +  k"x) 

=  2F{k,tan-^\/x).  [17] 

The  following  formulas  are  obtained  from  formulas  [10]- 
[13]  by  easy  substitutions  : 

r-.=^==  =  isn-Y^^-\-->.>/.>0;[19] 

a>lj>x>0;  [20] 

.r>/^>0;   [21] 

r°    ^  -^^  =ldn-Y^,  iZEZ'Y 

^^'^   y/(aF—x^(x^  —  P)      «  V«  «       y 

a>x>I>>0',  [22] 
f^  -^-  -    S-Y--    -^ZEi^Y  [23] 


CiiAi-.   XVI.]  ELLIPTIC    LNTEGllALS.  273 

For  example  we  will  take  [19]. 

Let  y  =  - '  then  dx^= '-  • 

^  ir 


Let  now  ,~  =  tn/  and 


Jo  V(i-ay)/       u'      \ 


r- (ijl _i  r^ (iz 

From  [l-iJ-ClT]  may  be  derived  in  like  manner 

-*'    V(x-a)(x-/3)(.f-y)       Va-y  V^-'— 7   ^""r, 

,r>a>^>y;    [24] 

f  ''■'  =-!=..-{  SE1.SE}\ 

^^     ^(a-j:)(x-l3)(x-y)        Va-y  V^a-/S    ^a-y/ 

a>^>/3;    [IT.] 

^y  ^/(n-j')(ft-j-)(,—  y)      Va-y         \^/i-y  ^«-y/ 

y3>./->y;    [I'O] 

^'     -J(a-.r){fS-.r)(y-J-)        Va-y  V^^--/-   ^a-y/ 

y>-'-;  [-''] 

For  example  we  will  take  [24]. 

Let   ?/  = »   tlien  dx  ^ ^  ; 

x  —  y  if 


274  INTEGRAL   CALCULUS.  [Aux.  204. 


X 


-J{x-a){x-p)ix-y) 

d,, 


^0     V//(l-(a 


y)y)(i-(/^-y)//) 

Let  now  z  =  {a  —  y)y,  then 
Jo  Jjj  n  —  /'^  —  .v^  / 


Vy(l-(a-y)y)(l-(^-y)2/) 


1 r-^ 

1  — ,,  »^ii     / 


dz 


Va  — y 
From  [24]-[27]  may  be  obtained 

s:- 


V(x  -  a)  (a;  -  )8)  (x  -  y)  (^  -  8) 
2 


V(a-y)(^-8) 

a->a;    [28] 

V(a  —  X)  (j 


(x-/3)(x-y)(:r-8) 
2 


/       /^__8      „_^     Jg-^       y-A 

V^a-/3    CC-8    ^a-y     fi-hj 


V(a-y)(^-S) 

a>r>y3;    [29] 


Jx   ■^(a-x){p-x){x-y){. 


(^-8) 

2 


sn" 


J\a-j,_^-x\p-y^a-^\ 


V(a-y)(/8-8) 

/3>.r>y;    [30] 


CiiAi-    \vi.]  ELLIPTIC  inti-:(;i:al8. 

9 


=  sn- Y  J^Zli  .  yjZ:!-,  J— /^ .  y  -  A. 
-8)  \^y-8    )8-^-    ^a-y    /i-dj 


V(a-y)(/3 

y>./->8;    [;!1] 
dx 


X 


V(a-.r)(y3-r)(y-a')(8--T) 


V(a  — y)(^— 8)  \>a  — 8    y  -  .>•      >' a  -  y     /^-8 

8>...    [;i2] 

Formulas  [i^4]-[32]  enable  us  to  integrate  the  reciprocal  of 
the  square  root  of  any  cubic  or  biquadratic  whicli  has  real 
roots. 

As  an  example  let  us  find    I  =• 

*^*J    V(2  ax  —  x^)  (a-  —  xr) 


,I.r 


V  (2  (I  —  X)  ('(  —  X)  X  (x  +  (i) 


V(2  a  —  x)  (a  —  X )  .'•  I  -'•  4-  //> 


(Ix 


-  r"-= 

•^a   V(2  a  —  a-)  («  —  a-)  a-  (j  +  ") 
=.l[sn-{l,f)-sn-(^4^)]bym 

_  1  sn-/'^,  ^^  -  1  //^,8ia->:^>\ 


276  INTEGRAL   CALCULUS.  [Art.  204. 

Formulas  [10]-[32]  suggest  the  appropriate  substitution  to 
rationalize  any  rational  function  of  x  and  the  square  root  of 
a  cubic  or  biquadratic  having  real  roots. 

For  instance,  let  us  consider    I     V(«"  —  x-)  {b-  —  x'^)  dx. 

Let  y  =  sn"'(  y,  -  j,  [v.  formula  [1<^]]. 

Then  a;  =  ^  sn  (  //,  j ,  dx  =^b  en  y  dn  //  dy,  iv^  —  x^=^  a^  dn^  y 
Ir  —  ./■-  =  1/  cu-  y  ; 

V(«^  —  X')  (U-  —  X-)  .  dx  =  (d>H     en-  y  dn-  y  dy 

Examples. 

(1)  Find    r  ,J^— •        Am.   ^  7vYmod  ^  V^  1-311- 

(2)  Kationalize  J    Vl  —  x* .  dx. 

A71S.  2  V2  jT  '((In^o-  -  dn*x)  dx.  fmod  ^\ 


X''  dx 

VC«^  -  hx)  ( 


V(«'  —  Ox)  {hx  —  x^) 

2         /      /A       2      /  h\ 

Ins.    -  sn"M  1,  -     or  -A'    mod  -    • 


(4)    Rationalize  J  J'l—Z]!!,  dx. 

Avs.   2<i    I      dn'^a- .  dx,  or  2a  Ki  -,  -  V 
J.  \n    1) 


Chap.   XVI.]  ELLIPTIC    INTEGRALS.  277 

dx 


(5)    Find  J^ 


Ans.   i^sn-(l,^^)oriA'(moa^^). 
Suggestion :  let  z=^—^. 

205.  If  we  are  integrating  the  reciprocal  of  the  square  root 
of  a  cubic  or  biquadratic  having  imaginary  roots,  formulas 
[24]- [32]  of  Art.  204  no  longer  serve  our  purpose  and  we 
are  driven  to  a  more  laborious  method. 

We  need  only  to  consider  the  biquadratic  form  as  we  may 
regard  the  cubic  as  a  special  case  under  it. 

dx 
V(a  -\-2bx-\-  cx^  (a  +  2  /3x  +  yx^) 

let  ?/  = where  m  and  n  are  at  first  undetermined.     "We 

x  —  n 

shall  get  an  integral  of  the  form 


Take  the   form    |  — ,  and 


/: 


V(^  +  %  +  C,f)  (.4'  +  B',j  +  CV) 

and  m  and  n  can  then  be  so  'chosen  that  B  and  B'  sliall  be 
equal  to  zero,  and  the  integral  can  be  obtained  by  one  of  the 
formulas  [18]-[21]  of  Art.  204. 

The  values  of  m  and  n  required  are  easily  proved  to  be  the 
roots  of  the  quadratic  equation 

ra  —  ny  Im  —  ii^ 


hy-r^'         hy-rii 


0,  (1) 


and    are    always    real    if   tlie    original    biquadratic    lias    any 
imaginary  roots. 

d.r 


For  example  let  us  find    i 


Vx(H-a^') 


278  INTEGIIAL   CALCULUS.  [Art.  206. 

Here  a  =  0,  b  =  ^,  r  =  0,  a  =  1,  ^  =  0,  y  =  1.     Our  auxil- 
iary equation  (1)  becomes  r:^  —1  =  0  and  gives  1  and  —  1  for 

x  —  1 
m  and  n.     Let,  then,  y  =  >  substitute  and  reduce  and 


J'**     ax      /-  r' 
0     V^+  0-2)  ~"  ^  V_  1  V(l  +  //- 


Ki-y^) 


-2V2  f  ^  -^-  -2cn-r0,f ) 

Jo  V(l  +  a;^)  (1  -  a:^)  \      2  ^ 

=  2  A' r mod  ^  J  =3.708. 


Examples. 

SiKjfjestlon  :   let  s  =:  x^. 

(2)  Rationalize  f '— =^^=. 

-^"    V.r(l+a;2) 

f'''l  —en?/ 
Jo     1  +  en  //     -^ 

r^'2-2cn?/  — sn'^?/    ,    /       ,  V2\ 

=  I      ■-. •  dif     mod  — -    • 

Jo  sn^y  ''\  2  J 

(3)  Rationalize  f   \l I -\- x*  ■  dx. 

Jo     (1  +CI1  y)2     "^ 

Jo  sn^r  \  2   / 

20G.    Formulas  for  integrating  sna*,  en  a-,  dna-,  and  their 
powers  positive  and  negative,  are  obtained  without  difficulty. 


Chap.    XVI.]  ELLIPTIC    INTEGRALS.  279 

/I    /•— A-sna-cna-r/j-  1    C       (li( 

s\\xdjc  =  —  —,  I   =  — T   I  — 
A"  J              en  X  k  J    yj  ,j-i  _  /.'» 


=  -l  log  (y  +  vy'  -  Z^-")  if  Z/  =  tin  or.     Hence 
I  su  xdx  =  —  -  log  (dn  a-  +  Vdn-  j  —  A-"*) 

=  -ic„s,..(iiH).  [,J 

I  en  .r  dx  =  -  cos~ ^  (du  x).  [2] 

I  dii  X  dx  =  am  a;  =  sin~^  (sn  a*).  [3] 

/dx  _  /•  sn  cr  en  a-  dn  a-  dx j  P                dij 
sn  a"     »/     sn'^  x  en  a*  dn  a:         "J  U'^J  (1  —  u)  (1 k'^ii) 

=_ ,  log  rV(i-y)(i-A-V)+i  _  L±i:1 


if  y  =  sn^  X. 
Hence 


Jtoi:  =  2l^'^"'    L di?^ J 

1  ^        ,  rA;'sna-  — on./"] 

=  — tan-M  t; i •  [6] 

A;'  LA-'sna-  +  c-n.rJ  •-   -• 

From  Art.  198  [1]  we  get 

f'sn^  X  dx  =  ].,  [.r  -  E  (am  .r.  A-)],  [7] 


280  INTEGRAL   CALCULUS.  [Art.  206. 

cn2  X  dx  =  -  \_E  (am  x,  k)  —  A-'^x],  [8] 

/    dn^  xdx=^  E  (am  x,  k) .  [9 ] 

An  important  set  of  reduction  formulas  by  which  the  integral 
of  any  whole  power  of  sn  x,  en  x,  or  dn  x  can  be  made  to 
depend  upon  the  formulas  just  obtained  can  be  found  with- 
out difiiculty. 

We  have  — -  (sn"'+ '  .r  en  x  dn  x) 

dx  ' 

=  {m  +  1)  ^iV'x—{m  -f  2)  (1  +  A;2)  sxr  +  '-x-\-{m  +  3)  7r  sn"'  +  -'  a-, 
whence  we  get 

(m.  +  1)1    ^n""  xdx 

=  {m  +  2)  (1  +  Z-2)J'  ^^^-^  +  ^xdx 

—  (m  +  3)  k'-J    sn'"  +  ■•  x  dx  -f  sn'"  + '  a;  en  a:  dn  x.       [10] 
(m  +  1)  k'  ■  j    en'"  X  dx 

=  {in  +  2)  (/.•'  =^  -  Z-^)  J"  en'"  +  '-xdx 
+  (m  +  3)  /.-^J    cn"'  +  ^a-  r/.r  —  cn'"  +  i  a-  sn  a;  dn  x.       [11] 
(/M  +1}  /.'- 1    dn'"a;(/a3 

=  (»i  +  2)  (1  +  k'  -)  J'  dn'"  +  2.r  rZx 

—  (/«  -I-  3)  J   dn'«  +  ■•  X  dx  +  /.•-  dn'"  + '  a-  sn  a;  en  x.       [12] 


CiiAi-.  XVI.]  ELLIPTIC    INTEGRALS.  281 

Examples. 

(1)    Obtain  the  following  formulas  : 


I    sn ~ '  X  (/x  ^  x  sn ~ '  a-  +  7  cosh "" '  (  /  '    ) 

j    cn-» xdx  =  xcn-^x  —  -  cos"' (V/.-'^  +  A-^x^ 
j   dn~^xf/x=  a;dn~'x  —  sin~M  ^ — ^  )• 

(2)  Yim\ab£    fl-^^^i^'sn^x  +  ^'sn^rlr/x. 

^n5.    i  ^  1^0^^  +  b')  E  [j^,  1^  -  (a-^  +  2  //)  A'  (muA  ^^^ 

V2    C"  f  V2\ 

(3)  Find  ~   J      (dn^  x  -  du^  x)  dx,   i  mod  -^  j  • 

V2      /  V2\ 

^ri5.   ^/if  mod  ^y- j.  or  0.211) 

^.    -     r"2-2cna--sn='a:  ,      /       ,  V2\ 

Find    /     -, dx,   I  mod  -—    • 

Ji)  sn^a;  \  2  / 

A'—  2  A'  r mod  ^  j-  or  O.aj;? 


(4) 


(5)  A  cannon-ball  of  radius  h  is  fired  horizontally  through 
the  middle  of  a  ship's  mast  (radius  a) ;  find  (a)  the  volume, 
and  (b)  the  whole  superficial  area  of  the  plug  required  to  fill 
tlie  hole. 

Ans.    (a)  •'^[(-  +  /'V^(''  ")-^'^  +  2/,^,A(mud^^)]; 
(b)  8(a  +  /.;^'A('^  !^)-("-/')A'(n..Ml''^jJ- 


282  INTEGRAL   CALCULUS.  [Akt.  206. 

(6)  A  cylindrical  hole  of  radius  h  is  bored  through  a  sphere 
of  radius  a  and  just  grazes  the  centre  ;  find  (a)  the  area  of  the 
inner  surface  of  tlie  hole,  (b)  the  s}3herical  surface  removed, 
and  (c)  the  spherical  volume  removed. 

Ans.    (a)  A:ab  e{  -'?-); 

(b)  2aV-4«.^^0\  l); 
(c)f7ra3-4,3^  a-        ^^2; 

(7)  Find  the  mean  distance  of  points  uniformly  distributed 
along  the  perimeter  of  an  ellipse  from  a  focus. 

Aiis.    One  half  of  the  major  axis. 


UlAP.  XVll.J  TIlEUliV    OF    FUNUTIUNS.  283 


CTTAPTER    XVII. 

INTItOn'uCTIOX    TO    THE    THE(MIY    OK    FUNCTIONS. 

207.  A  function  having  but  a  single  value  for  any  given 
value,  real  or  imaginary,  of  the  variable  is  called  a  simjle-valued 
function.  Rational  Algebraic  Functions,  Exponential  Func- 
tions, the  direct  Trigonometric  Functions,  and  the  Kllii)tic 
Functions  are  single-valued. 

A  function  which  has  in  general  two  or  more  values  for  any 
given  value  of  the  variable  is  called  a  mnltiple-vahied  function. 
Irrational  Algebraic  Functions,  Logarithmic  Functions,  tiie 
inverse  or  anti-Trigonometric  Functions,  and  the  Elliptic  In- 
tegrals, are  multiple-valued. 

208.  In  Chapter  II.  we  have  explained  the  customary  graph- 
ical method  of  representing  an  imaginary  by  the  position  of  a 
point  in  a  plane,  the  rectangular  coordinates  of  the  point  being 
the  real  term  and  the  real  coeflicient  of  the  pure  imaginary  term 
of  the  imaginary  in  question. 

In  the  ordinary  treatment  of  tlie  Tlioory  of  Functions  this 
metiiod  of  representation  is  of  the  greatest  service,  and  enables 
us  to  bring  the  study  of  functions  of  imaginary  variables  within 
the  province  of  Pure  Geometry,  and  to  give  it  great  deliniteuess 
and  precision. 

For  the  sake  of  brevity  we  shall  in  future  use  the  symbol  / 
for  V  — 1  and  cis(^  for  cos<^  -f-  V—  1  sin<^,  so  that  we  shall 
write  our  ty|)ical  imaginary  as  x  +  yi  or  as  rcis<^,  instead  of 
using  the  longer  forms  .;•  -f  »/  V  —  1 ,  and  r  (cos </>-(-  V— 1  sln*^). 

We  shall  also  use  the  name  complex  f/uautit;/  for  an  iniaginarv 
of  the  typical  form  when  it  is  necessary  to  distinguish  it  from 
fi  pure  imaginary. 


284 


INTEGRAL  CALCULUS. 


[Art.  209. 


209.  A  complex,  variable  z  =  x-\-yi  is  said  to  vary  co//////»- 
OMsly  when  it  varies  in  such  a  manner  that  the  path  traced  by 
the  point  (.T,y)  representing  it  is  a  continuous  line. 


Thus  if  z  changes  from  the  value  a  to  the  value  y8,  so  that 
the  point  representing  it  traces  any  of  the  four  lines  in  the 
figure,  z  varies  continuously. 

It  will  be  seen  that  a  variable  can  pass  from  the  first  to  the 
second  of  two  given  values,  real  or  imaginary,  by  any  one  of 
an  infinite  number  of  different  paths  without  discontinuity  if  the 
variable  in  question  is  not  restricted  to  real  values  ;  while  a  real 
variable  can  change  continuously  from  one  given  value  to  another 
in  but  one  way,  since  the  point  representing  it  is  confined  in  its 
motion  to  the  axis  of  reals. 


210.  A  single-valued  function  w  of  a  complex  variable  z  is 
called  a  continuous  function  if  the  point  representing  it  traces 
a  continuous  path  whenever  the  point  representing  z  traces  a 
continuous  path. 

A  multiple-valued  function  of  z  is  continuous  if  each  of  the  n 
points  representing  values  corresponding  to  a  value  of  z  traces 
a  continuous  path  whenever  z  traces  a  continuous  path. 
These  n  paths  are  in  general  distinct,  but  two  or  more 
of  them  will  intersect  whenever  z  passes  through  a  value 
for  which  two  or  more  of  the  n  values  of  w,  usually  distinct, 
happen  to  coincide.     Such  a  value  of  z  is  sometimes  called  a 


CiiAP.  XVII] 


THEORY    OF    FUNX'TIONS. 


285 


critical  value,  and  the  consideration  of  critical  valiu'.s  plays  an 
important  part  in  the  Theory  of  Functions. 

In  studying  a  nniltiple-valued  function  we  may  confine  our 
attention  to  any  one  of  its  n  values,  and  except  for  the  possible 
presence  of  critical  points  this  value  may  be  treated  just  as  we 
treat  a  single-valued  function. 

In  representing  graphically  the  changes  produced  in  a  func- 
tion tv  by  changing  the  variable  z  on  which  it  depends,  it  is 
customary  to  avoid  confusion  by  using  separate  sets  of  axes  for 
w  and  z. 


211.  If  we  use  the  word  function  in  its  widest  sense, 
w  =  it  +  vi  will  be  a  function  of  a  complex  variable  z  =  x-\-  v«, 
if  u  and  v  are  any  given  functions  of  x  and  y.     For  example, 


6?/,  .^-  +  ?/^    x-yi,   x^ -y' +  2xyi, 


X  —  y  -^-  xt 


Var^  +  y-  +  4 
may  all  be  regarded  as  functions  of  z. 

We  have  seen  in  Chapter  II.,  Arts.  .36-42,  that  with  tliis 
definition  of  function  the  derivative  with  respect  to  z  of  a  func- 
tion IV  is  in  general  indeterminate  ;  but  that  there  are  various 
functions  of  :,  for  instance,  z"*,  logz,  e',  s'mz,  where  the  deriva- 
tive is  not  indeterminate.  We  are  now  ready  to  investigate 
more  in  detail  the  general  question  of  the  existence  of  a  deter- 
minate derivative  of  a  function  of  a  complex  variable. 

Let  ?o  =  M  +  vi  be  a  function  of  2 ;  u  and  v,  which  are  real, 
being  functions  of  x  and  y. 

Starting  with  the  value  Zo  =  x^)-\-y^,i  of  z  and  the  correspond- 
ing value  Wo  =  "o  +  ^'ui  of  w,  let  us  change  z  by  giving  to  x 
increment  Ax  without  changing  y. 


Let  A^'<  and  A^v  be  the  corresponding  increments  of  «  and 
V ;  and  2i  and  Vi  the  new  values  of  :;  ami  iv. 


286 

We  have 
Tlieii 


INTEORAL   CALCULUS. 


[Art.  211. 


^0+  -•'•' 


v\,  +  Aj«  +  I'A^V. 


^x         Ax 


and  the  derivative  of  "•  with  respect  to  z  under  the  given  cir- 
cumstances is 


^~=o[':;::]=--'^^^- 


[1] 


■To 


If,  however,  starting  with  tlie  same  value  z;„  of  z,  we  change 
z  by  giving  ^  the  increment  Ay  without  changing  x,  we  have 

;2i  =  A,  +  (.'/.  +  A.v)  /  =  z„  +  /A?/, 


and 


W,  -  Wo 

AyM       lAyV 
~  iA//"*"  /A?/ 

Zj  -  Zo 

limit 

~"'i-^^'"1_/) 

A2=0 

z,  —  z,.              * 

[2] 


and  this  is  tiie  derivative  of  w  with  respect  to  z  when  we  cluinge 
y  and  do  not  change  a;. 

Comparing  [1]  with  [2],  we  see  that  if  we  start  with  a  given 
value  of  z,  and  change  z  in  the  two  different  ways  just  con- 
sidered, the  limits  of  the  ratios  of  the  corresponding  changes  in 
w  to  the  changes  in  z  need  not  be  the  same.     Indeed,  the  two 

values  for  —  given  in  [1]  and  [2]  will  not  be  the  same  unless 

to  =  11+  'vi  is  sucli  a  function  of  z  =  x  +  yi  that 

J)^n  =  I\r     Mild      Il,„  =  ~D^r.  [3] 


CllAl'.  XVII.]  THKOllV    OF    FINCTKJNS.  2S7 

We  shall  now  show  that  if  w  is  such  :i  function  of  z  that 

tlio  same  if  we 


equations  [3]  are  satisiied,    v-^^o    —    ^^''^  ^'*^ 

start  with  a  given  value  z^y  of  z,  no  uiatttcr  in  what  manner  z 
may  change  ;  that  is,  no  matter  in  what  direction  the  point 
representing  z  may  be  supposed  to  move  ;  or.  in  other  words, 

no  matter  what  may  be  the  value  of    .     .  ,v  r 

We  have  in  general,  since  w  is  a  function  of  the  two  variables 
X  and  y, 

\iv  =  (D,  H  +  ilJ,  r)  A.I-  +  ( I)^  n  +  iD^  r)  A//  +  t, 

where  e  is  an  infinitesimal  of  higher  order  than  Ax-  or  A//. 

(I.,  Art.  Ut8.) 
A2  =  A.r  +  i\y. 


Hence 


\w_D^u.\x+  iD^ v.\y-\- iD, v .  A.r  +D^u.:^y-\- (. 
Az  Ax  +  iAy 

A.'-  A.r      Ax 


Ax 


limit 

AZ: 


nit  rA?<.'~|  _  dw 
=  ^l^]~  dz 

alue  involving    a'"^o       "   '  -'"'^  therefore  dei)endent  ui)on 


the  direction  in  which  z  is  made  to  move. 
If,  however,  [3]  is  satisfied,  [4]  reduces  to 


ilz 


and  the  derivative  of  ?'.'  is  independent  af  ^^ 


nit    fAvl 

^<>La:J 


288  INTEGRAL  CALCULUS.  [Art.  212. 

A  function  which  satisfios  equations  [3],  and  which,  there- 
fore, has  a  derivative  whose  value  depends  only  upon  the  vahie 
of  the  independent  variable,  and  not  upon  tlie  direction  in  which 
the  point  representing  the  variable  is  supposed  to  move,  is  called 
by  some  writers  a  monoijenic  finictiou,  by  others  a.  function  ichlrh 
has  a  derivative. 

212.  Any  function  of  n  which  can  be  formed  by  performing 
an  analytic  operation  or  series  of  operations  upon  «  as  a 
whole,  without  introducing  x  and  i/  except  as  they  occur  in  z, 
is  a  monogenic  function  of  z. 

Yor  if  w  =fz  =f(x  4-  yi) , 

where /2  can  be  formed  by  operating  upon  z  as  a  whole, 

D^iv=f'z,  and  Dytv  =if'z\ 

therefore   iD^ic  =  Dyio,       or    iD^  (u  +  -^0  =  D^  {n  +  tri)  ; 

wlience         D^n  =  D^v,         and  DyU  =  —  D^v\ 

and  [3],  Art.  211,  is  satisfied.     Consequently  w  is  monogenic. 
This  accouuts  for  the  results  of  Arts.  38-42. 

If  w  is  a  multiple-valued  function  of  z,  there  may  be  several 

diflFerent  values  of  — ,  corresponding  to  the  same  value  of  z ; 

dz 
but  if  'W  is  uionogenic,  each  of  these  values  depends  only  upon  z.- 
and  not  upon  the  way  in  which  z  is  supposed  to  change. 

In  future,  unless  something  is  said  to  the  contrary,  we  shall 
give  the  name  function  only  to  monogenic  functions.  Thus  we 
sludl  not  call  such  expressions  as  x  —  yi,  or  si?  +  y^  +  2xyiy 
functions  of  z. 

Conjugate  Functions. 

213.  If  n  and  v  are  functions  of  x  and  y,  satisfying  equations 
[3],  \\{.  211,  it  is  easy  to  prove  that 

D^^ u  +  D^^u  =  Q        and        D^ v  -f  Z>>  =  0- 


Chap.  XVll.]  THEOIIY   OF    FUNHrriONS.  289 

For  since  D^  u  =  I)^  r  :iik1  IJ^  c  =  —  D^ii, 

we  have  D/ u  =  D^D^v  ami         D;h  =  -  nj),v, 

D/  V  =  -  I),  I)^  n       and  /r-  /•  =  1>^  I)^  „  ; 

u  and  V  are  then  sohitions  of  I>aphice's  equation, 

I)/V+D;V=Q.  [1] 

Any  two  functions  <^  and  ^  oi  x  and  y,  such  that 
ff){x,y)  +  ill;  {x,y)  is  a  niouogenic  function  of  x  +  yi,  are 
called  conjugate  functions  ;  and,  by  what  has  just  been  proved, 
each  of  a  pair  of  conjugate  functions  is  always  a  solution  of 
Laplace's  Equation  [1]. 

Thus  ar  —  y-,    2xy;  e'cosy,  e^s\u>/\    ^  log  (.x- +  ^-),  tan  '•''; 

X 

are    three    pairs  of   conjugate    fuuftions,   since   x-  —  y--{-2xyi 

=  {x  -{•  yiy,  e*  cosy  +  ie'  sin y  =  6'+»"',  -i  log  (x^  +  y')  +  i  tan~*^ 

=  log  {x  +  yi) ,  and  consequently,  by  Art.  212,  are  all  monogenic. 
Therefore  each  of  the  six  functions  at  the  beginning  of  this 
paragraph  is  a  solution  of  Laplace's  Equation  [1]. 

It  is  clear  that  we  can  form  pairs  of  conjugate  functions  at 
pleasure  by  merely  forming  functions  of  x  +  yi  and  breaking 
them  up  into  their  real  parts,  and  their  pure  imaginary  parts  ; 
that  is,  throwing  them  into  the  typical  form  u  -\-  ri. 

If  each  of  a  pair  of  conjugate  functions,  <^  and  tp,  is  written 
equal  to  a  constant,  the  equations  thus  formed  will  represent  a 
pair  of  curves  which  intersect  at  right  angles.  For  let  (.r,  y) 
be  a  point  of  intersection  of  the  curves  <f>  =  a,  ip  =  b  ;  the  slopes 

of  the  two  curves  at  (.r,  y)  are  respectivelv  —  — ^,  —  — ^  by 

I.,  Art.  202;  and  since  D^4>=D^\p  and  iJ^ij/ =  —  D^<f>,  the 
second  slope  is  minus  the  recii)rocal  of  the  first,  and  the  curves 
are  perpendicular  to  each  other  at  the  point  in  question. 

Thus  or  —  v/2  =  a,  2xy=  h,  cut  each  other  orthogonally  ;  as  do 


290  LNTECJKAL   CALCULUS.  [Art.  214. 

also  ^\og{x'  +  y-)=a,  tan  ^•'  =  h\    or,  what   amounts   to  the 

same   thing,   ar  +  ir  =  a^  ^  =  hi.      It   must   be  observed,  how- 
o; 

ever,  that  ar  +  y-  and  -  are  not  conjugate  functions,  and  that 

X 

in  general  the  converse  of  our  proposition  does  not  hold. 

It  may  be  easily  proved  that  if  <t>  and  ip  are  conjugate  func- 
tions of  X  and  y,  and  /  and  F  are  any  second  pair  of  conjugate 
functions  of  a;  and  y,  the  new  pair  of  functions  formed  by  re- 
placing X  and  y  in  <^  and  ij/  by  /  and  F  respectively  will  be 
conjugate. 

Thus  (e'cosjf/)-  — (e'sin//)-,  2e'cos?/.e'sin2/, 

or,  reducing,  e^cos2y,  e^s\n2y, 

are  conjugate  functions  ; 

^log  [(o;^-  fr  +  V2xyy-],  tan-'^-^), 

or,  reducing,  log  (x-  +  //-') .   tan  '  ( -^^X 

are  conjugate. 

The  properties  of  conjugate  functions  given  in  this  article 
are  of  great  importance  in  many  branches  of  Matliematical 
Physics. 

Example. 

Show  that  if  x'  and  y'  are  conjugate  functions  of  x  and  y, 
X  and  y  are  conjugate  functions  of  x'  and  y'. 

l^reacrvation  of  Anr/lcs. 

214.  If  w  is  a  single-valued' monogenic  function  of  z,  and 
the  point  rei)resenting  z  traces  two  arcs  intersecting  at  a  given 
angle,  the  corresponding  arcs  traced  by  the  point  representing 
tc  will  in  general  intersect  at  the  same  angle. 


CiiAr.  XVII.] 


THEORY   OF   FUNCTIONS. 


For  let  Zq  be  the  point  of  intersection  of  the  cnrves  in  the  2 
plane,  and  ?'•„  the  corrosponding  point  in  tlie  it-  plane.  Let  2,  be 
a  point  on  the  tirst  curve,  and  z.,  a  point  on  the  second  ;  and  let 


^9<=:^^, 


Wi  and  lOo  be  the  corresponding  points  in  the  ?o  figure. 

Let  ri,  /'a,  Si,  and  s.^  be  the  moduli  of  «,  —  Zq,  Zo  —  Zq^  ju,  —  ?fo, 
and  W2  — 10^  respectively,  <^i,  <^o,  i/^i,  and  1/^2  their  arguments : 
then,  since  to  is  a  monogenic  function  of  z,  we  must  have 


lim 


it  r!!i^i^i= limit  r^^viii^i 
limit  r^ii^n= limit  r^^^i^^i; 

\_i\  cis  ^ij  |_?-2  cis  <^2  J 


whence,  by  Art.  23, 

limit    —cis  (t/'i  —  <^i)    =  limit    -  cis  (i/^^  —  <^2)    » 

and  since,  when  two  imaginarit's  are  equal,  tlicir  moduli  must 
be  equal,  and  their  arguments  must  be  equal,  unless  the  moduli 
are  both  zero  or  both  inlinite, 

limit  {\p.,  —  i/r,)  =  limit  (<^2  —  </>,)  ; 

that  is,  the  angle  between  the  arcs  in  the  in  figure  is  equal  to 
the  angle  between  the  corresponding  arcs  in  the  2  figure  ;  unless 


[g..="'  -  m.-.: 


If  10  is  a  multiple-valued  monogenic  fimction   of  2,   and   if 
Starting  from  any  point  2o,  the  [joint  which  represents  z  traces 


292  LNTEGUAL  CALCULUS.  [Art.  215, 

out  two  curves  intersecting  at  an  angle  a,  each  of  the  n  points 
representing  the  corresponding  values  of  to  will  trace  out  a  pair 
of  curves  intersecting  at  the  angle  a ;  unless  Zq  is  a  point  at 

which  —  is  zero  or  infinite. 
dz 
If,  then,  w  is  any  monogenic  function  of  z,  and  the  point 
representing  z  is  made  to  trace  out  any  figure  however  complex, 
the  point  rei)resenting  w  will  trace  out  a  figure  in  which  all  the 
angles  occurring  in  the  z  figure  are  preserved  unchanged,  except 
those  having  their  vertices  at  points  representing  values  of  z 

which  make  —  zero  or  infinite. 
dz 

This  principle  leads  to  the  following  working  rule  for  trans- 
forming any  given  figure  into  another,  in  which  the  angles  are 
preserved  unchanged. 

Substitute  x'  and  y'  for  x  and  y  in  the  equations  of  the  curves 
which  compose  the  given  figure,  x'  and  j'  being  any  pair  of 
conjugate  functions  (Art.  213)  of  x  and  y,  and  the  new 
equations  thus  obtained  will  represent  a  set  of  curves  forming 
a  second  figure  in  which  all  the  angles  of  the  given  figure  are 
preserved  unchanged,  except  those  having  their  vertices  at 
points  at  which  D^x'  and  D^y'  are  both  zero,  or  at  which  one  of 
them  is  infinite. 

For  exniiiple,  x  —  y  =  a,  (1) 

x+y=b,  (2) 

are  a  pair  of  perpendicular  right  lines.  Replace  x  by  xr  —  y' 
and  y  by  2  xy,  and  we  get 

aJ_2a'j/-?/2  =  a,  (3) 

a^+2.T?/-y-  =  6,  (4) 

a  pair  of  hyi)crl)<)las  that  cut  orthogonally. 

215.  If  ?o  is  a  sinc/le-vcdued  continuous  function  of  z,  it  is 
clear  th.it  if  Wq  and  v\  are  the  values  corresponding  to  z„  and  Z], 


CiLiP.  XVII.]  Til EOUV   OF   FUNCTIONS.  293 

aud  the  point  z  moves  from  Zq  to  rj  by  two  different  paths,  the 
corresponding  paths  traced  by  xv)  will  begin  at  ii\  and  end  at  jc,, 
and  consequently  that  if  z  describes  any  closed  contour,  w  also 
will  describe  a  closed  contour. 

If  ?c  is  a  double-valued  function  of  «,  since  to  each  value  of 
z  there  will  correspond  two  values  of  w,  it  is  conceivable  that 
if  n.\  and  Wx  are  the  values  of  ?o  corresponding  to  ^i,  and  z  moves 
from  2o  to  2j  by  two  different  paths,  v:;  may  in  one  case  move 
from  Wq  to  t(7i,  and  in  the  other  case  from  rv,  to  w^ . 

It  can  be  proved,  however,  that  if  the  two  paths  traced  by  z 
do  not  enclose  a  critical  j^oint  (Art.  210),  and  w  is  finite  and 
continuous  for  the  portion  of  the  plane  considered,  tliis  will 
not  t  ike  place,  and  that  the  two  paths  starting  from  Wf^  will 
terminate  at  the  same  point  tvi.  We  give  a  proof  for  the  case 
where  ;s  is  a  single-valued  function  of  iv. 

As  z  traces  the  first  path,  each  of  the  two  points  repre- 
senting the  two  values  of  lo  will  trace  a  path,  one  starting  at  icq, 
and  the  other  at  tco,  and  unless  the  z  path  passes  through  a 
critical  point,  the  two  tv  paths  will  not  intersect,  but  will  be 
entirely  separate  and  distinct,  and  will  lead,  one  from  ivq  to  icx, 
the  other  from  iVo  to  tVi'. 

If,  now,  the  z  path  be  gradually  swung  into  a  second  position 
without  changing  its  beginning  or  its  end,  since  w  is  a  continu- 
ous function,  the  two  tv  paths  will  be  gradually  swung  into  new 
positions  ;  but,  provided  that  the  z  path  in  its  changing  does  not 
at  any  time  pass  through  a  critical  point,  the  two  w  paths  will 
at  no  time  intersect,  and  consequently  it  will  be  impossible  for 
the  w  points  to  pass  over  from  one  path  to  tiie  other,  and  there- 
fore the  point  which  starts  at  «•„  must  always  come  out  at  tf,, 
and  not  at  ic/. 

It  f()llows  readily  from  this  reasoning  that  if  z  describes  a 
closed  contour  not  embracing  a  critical  point,  each  of  the  w 
points  will  describe  a  closed  contour,  and  tiiese  contours  will 
not  intersect. 

Of  course,  the  proof  given  above  holds  for  any  nuiltiplc- 
valued  function. 

In    any  portion  of  the    plane,  then,  not  containing   critical 


294 


INTEGliAL    CALCULUS. 


[Art.  210. 


points  the  separate  values  of  a  multiple-valued  function  may  be 
separately  considered,  and  may  be  regarded  and  treated  as 
single-valued  functions. 

216.  That  in  the  case  of  a  double-valued  function  two  paths 
in  the  z  plane,  including  between  them  a  critical  point,  but 
having  the  same  beginning  and  the  same  end,  may  lead  to 
different  values  of  the  function,  is  easily  shown  by  an  example. 

Let  w  =  Vz,  and  let  z,  starting  with  the  value  1 ,  move  to  the 
value  —  1  by  the  semi-circular  path  in  the  figure.     That  one  of 


the  corresponding  values  of  v\  winch  starts  with   +  1  will  de- 
scribe the  quadrant   shown    in  the  figure,  and  will   reach  the 


point  1  .cis-, 


If,  however,  z  moves  from  +1  to  —  1  by 


the  semi-circular  path  in  the  second  figure,  the  value  of  w  which 
starts  with  + 1  will  describe  the  quadrant  shown  in  the  second 

figure,  and  will  reach  the  value  l.cisf— ^J,  or  —i.      These 

two  paths  described  by  z^  then,  although  beginning  at  the  same 
point  -{■  1  and  ending  at  tlie  same  point  —  1,  cause  that  value 
of  the  function  which  begins  with  +  1  to  reach  two  different 
values ;  and  the  two  paths  in  question  embrace  the  point  3  =  0, 
which  is  clearly  a  point  at  which  the  two  values  of  jo,  ordinarily 
different,  coincide  ;  that  is,  a  critical  point. 


Chap.  XVII.] 


THEOnV    OF    FUNCTIONS. 


OOi 


It  is  easily  scon  that  if  z,  starting  witli  the  vahie  +1,  de- 
scribes a  complete  circumference  about  the  origin,  the  value  of 
w  which  starts  from  the  point  +  1  will  not  describe  a  closed 
contour,  but  will  move  through  a  semi-circumference  and  end 
with  the  point  I.cIstt  or   —1.     Now,    by  Art.  215  any   path 


described  by  z  beginning  with  +  1  and  ending  with  —  1  and 
passing  above  the  origin,  since  it  can  be  deformed  into  the 
semi-circumference  of  Fig.  1  without  passing  through  a  critical 
point,  will  cause  the  value  of  w  beginning  with  +  1  to  end  with 
+  i ;  and  any  path  described  by  z  beginning  with  -{- 1  and  end- 
ing with  —  1  and  passing  below  the  origin,  since  it  can  be 
deformed  into  the  semi-circumference  of  Fig.  2  without  passing 
through  a  critical  point,  will  cause  the  value  of  tv  beginning 
with  -f  1  to  end  with  —  i.  Therefore  any  two  paths  described 
by  z  beginning  with  -fl  and  ending  with  —1  will,  if  they  include 
the  critical  point  z  =  0  between  them,  lead  to  diflferent  values 
of  IV,  provided  that  the  same  value  of  lo  is  taken  at  the  start. 

217.  If  vj  is  a  douldc-valuod  function  of  z,  and  z  describes  a 
closed  contour  about  a  single  critical  point,  this  contour  may  be 
deformed  into  a  circle  about  tiie  critical  point,  and  a  line  lead- 
ing from  the  starting  point  to  the  circumference 
of  the  circle,  without  afTecting  the  final  value  of 
w  (Art.  215).  Thus,  in  the  figure,  the  two 
paths  ABCDA,  AB'C'D'B'A  lead  from  the 
same  initial  to  the  same  final  vahu-  of  "• ;  and 
this  is  true  no  matter  how  small  the  radius  of 
the  circle  B'C'D'. 


296 


INTEGRAL   CALCULUS. 


[Akt.  211 


Let  z<^  be  the  critical  point,  and  let  ir,,  be  the  corresponding 
point  in  the  w  figure.    As  z  moves  from  z^  towards  Zq,  the  points 


representing  the  corresponding  values  of  w  will  start  at  Wj  and 
iL\'  and  move  towards  w^,  tracing  distinct  paths. 

If,  now,  z  describes  a  circumference  about  z^,  and  then 
returns  along  its  original  path  to  Zi,  the  first  value  of  w  will 
either  make  a  complete  revolution  about  ioq  and  return  along 
the  branch  (1)  to  its  initial  value  ^^'l,  or  it  will  describe  about 


tu,,  a  path  ending  with  the  branch  (2)  of  the  w  curve,  and  move 
along  that  branch  to  the  value  w,'. 

In  the  first  case,  and  in  that  case  only,  the  value  of  w 
describes  a  closed  contour  when  z  describes  a  closed  contour, 
and  is  practically  a  single-valued  function. 

If   2(1  is   a  point  at  which  —    is  neither  zero  nor   infinite 
dz 

{y.  Art.  214),  when  z  describes  about  z„  a  circle  of  infinitesimal 
radius,  to  will  make  about  ?<•„  a  complete  revolution  ;  for  since 
if  two  radii  are  drawn  from  Zq,  the  curves  corresponding  to  them 
will  form  at  ?''o  an  angle  equal  to  the  angle  between  the  radii, 
when  a  radius  drawn  to  tlie  moving  point  which  is  describing 
tlu'  circle  about  Zy  revolves  thiough  an  angle  of  3GU°,  the  cor- 


Chap.  XVII.]  THEORY   OF   FUNCTIONS.  297 

iesi)uudiug  liue  joiniug  h'o  with  the  moving  point  representing  lo 
will  revolve  throngh  360°,  and  we  shall  have  what  we  have 
called  Case  I. 

If,  then,  we  avoid  the  points  at  which  —  is  zero  or  infinite, 

we  shall  avoid  all  critical  points  that  can  vitiate  the  results 
obtained  by  treating  our  double-valued  or  multiple-valued  func- 
tions as  we  treat  single-valued  functions. 

A  critical  iwint  of  such  a  character  that  when  z  describes  a 
closed  contour  about  it  the  corresponding  path  traced  by  any 
one  of  the  values  of  lo  is  not  closed,  we  shall  call  a  branch  point. 

When  a  function  is  finite,  continuous,  and  single-valued  for 
all  values  of  z  lying  in  a  given  portion  of  the  z  plane,  or  when 
if  multiple-valued  it  is  finite  and  continuous,  and  has  no  branch 
points  in  the  portion  of  the  plane  in  question,  it  is  said  to  be 
holomorphic  in  that  portion  of  the  plane. 

Definite  Integrals. 

fz.dz  was  defined  in 
Art.  80  in  effect  as  follows  : 


X 


>  .  dz  =  J'^'*^  [fz,  {z,  -  zo)  +fz,  (z,  -  z,)  +  fz,{z,  -  z,)  +  - 


where  z„  ^g?  2^3, . . .  z„_i  are  values  of  z  dividing  the  interval 
between  Zq  and  Z  into  ?i  parts,  each  of  which  is  made  to 
approach  zero  as  its  limit  as  n  is  indefinitely  increased. 

is  the  line  integral  of  fz  (Art.  IH.'i)  taken 

along  the  straight  line,  joining  2,,  and  Z  if  Zg  :""!  ^  are  repre- 
sented as  in  the  Calculus  of  Iniaginaries. 

It  has  been  proved  that  if /z  is  finite  and  continuous  between 
Zq  and  Z,  this  integral  dipeiids  nuMely  upon  the  initial  and  final 
values  of  z,  and  is  equal  to  FZ-Fzo  where  Fz  is  the  indefiuite 
integral  \  fz.dz. 


298  INTEGRAL  CALCULUS.  [Art.  218. 

If  2  is  a  complex  variuble,  and  passes  from  Zoto  Z  along  any 

given  path,  we  shall  still  define  the  definite  integral  |    fz.clz  by 

[1]  where  now  z,,  Zg,  z^,  •••z„^i  are  points  in  the  j^iven  path. 

Two  important  results  follow  immediately  from  this  defini- 
tion : 

1  St.  That  (""fz .dz=~  C  fz . dz,  [2] 

if  z  ti'averses  in  each  integral  the  same  path  connecting  z,j  and  Z. 

2d.  That  the  modulus  of    I    fz.dz  is  not  greater  than  the 

line-integral  of  the  modulus  of  fz  taken  along  the  given  path 
joining  z^t  and  Z. 
If  we  let 

fz  —  IV  =  u-\-  vi,  z  =  X  -{-  yi,  u  =  <f>  (x,  y) ,  and  v  =  if;  {x,  y) , 
then  I    fz  .dz—  |  (  u-\-  vi)  (dx  +  idy) 

=j<f>  {^,  y)  (J-^'  +  'J  "A  {^,  y)  f'-*-'  -  J  "A  (^N  y)  '^y  +  'J  *^  (^''  ?/)  ^y^ 

[3] 
each  of  the  integrals  in  the  last  member  ])eing  the  line-integral 
of  a  real  function  of  real  variables,  taken  along  the  given  path 
connecting  ^o  and  Z. 

If  the  given  path  is  changed,  each  of  the  integrals  in  the 
last  member  of  [3]  will  in  general  change,  and  the  value  of 

I    fz .  dz  will  change  ;  and,  since  z  may  pass  from  Zq  to  Z  by  an 

infinite  number  of  different  paths,  we  have  no  reason  to  expect 

that   I    /z.fZz  will  in  general  be  determinate. 

We  shall,  however,  i)rove  that  in  a  large  and  ini[)ortant  class 

fz.dz  is  determinate,  and   de|)ends  for  its  value 

upon  Zn  and  Z,  and  not  at  all  upon    the  nature  of   the  path 
traversed  bv  z  in  <>;oin<>-  from  2,,  to  Z. 


CiiAP.  XVII.]  TIIEOUY   OV   FUNCTIONS.  290 

-I'J.     li  fz  is  holomorphic  in  :i  <rivtn  portion  of  the  pl.ine. 


S>-'~- 


[1] 

if   z    describes   any  closed    contour    lying   wholly    within  that 
portion  of  the  plane. 

From  [;5],  Art.  2iri.  wo  have 

j    \fz .  dz  =  iiv.  (h  =  j  ndx  +  /  j  vdx  —  Cvdij  +  /  Cmh/,     [2] 

the    integral    in    each   case  being  tlie   line-integral   aiounil   t!ie 
closed  contour  iu  question. 

Since  w  z=  fz  is  holomorphic,  u  =  <fi(x,  y),  and  v  =  i/^  (.r,  »/), 
and  D^u,  DyU,  D^v,  and  D^v  are  easily  seen  to  be  finite,  con- 
tinuous, and  single-valued  in  th^portionof  the  plane  considered. 
Therefore,  by  Art.  170, 

Cudx  =  f  (  D^ndxdy  ;        i  vdx  =  f  C D^vdxdy  ; 
Cvdy  =  -  r  Cn.i'dxdy  ;    fndy  =  -  j'  fl)^>,dxd;/ ; 

the  integral  in  the  first  nieniber  of  each  equation  l)eing  taken 
around  the  contour,  and  that  in  tiie  second   member  being  a 
surface-integral  taken  over  the  surface  bouuded  by  the  contour. 
We  have,  then,  from  [2], 

C'fz.dz  =  f  C(D^n  +  I)A')dxdy-\-  i  f  f(D^r-I)^H)dxdy,     [3] 

but  D,u  =  DyV,  and  D^u  =  -  D,c  from  [.}],  Art.  211.    Tlierefore, 
[3]  reduces  to  j    "fz .  dz  =  0. 

From  this  result  we  get  easily  the  verv  important  fact  that  if 

fz.dz  will 

have  the  same  value  for  all  paths  leading  from  z„  to  %,  provided 

they  lie  wholly  in  the  given  part  of  the 

plane.     For  let  z^/iZ  and  z^hZ  be  any 

two   paths    not   intesecting   between  r„ 

and  Z.     Then  z^aZbz^  is  a  closed  con-    ^ 

tour,  and 


300 


INTEGRAL   CALCULUS. 


[Akt.  220 


I    fz.dz  (along  ^o^'^^^^o) 
=  r  fz.dz  (along  z„aZ)  -f   C'fz.dz  (along  Zhz^) ■=()', 

but  I   "fz.dz  {aXong  Zhz^))  =  —  j    fz.dz  (along  2!„6Z) 

by  Art.  218. 
Therefore,   |    /^.r/^  (along  Zq^-Z^)  =  I    /^.  f/2;  (along  z^ft.^). 

If  the  paths  ZqCiZ  and  z^ftZ  inter- 
sect, a  third  path  z^^^cZ  may  be  drawn 
not  intersecting  either  of  them,  and 
liy  the  proof  just  given 

Jfz .  dz  (along  ZuaZ)=  i    fz.dz  (along  z^cZ) , 

I    fz.dz  (along  2oi-^)  =  j    fz.dz  (along  z„f-^)  ; 
therefore, 

X  fz.dz  (along  Zo^-^)  =  )    fz-dz  (along  ZuftZ). 

220.  If  fz,  while  in  other  respects  holomorphic  in  a  given 
portion  of  the  plane,  becomes  infinite  for  a  value  T  of  z,  then 
i  fz.dz  taken  around  a  closed  contour  embracing  T,  while  not 
zero,  is,  however,  equal  to  the  integral  taken  around  any  other 
closed  path  surrounding  T. 

For  let  ABCD  be  any  closed  con- 
tour about  T.  With  T  as  a  centre, 
and  a  radius  e,  describe  a  circumfer- 
ence, taking  e  so  small  that  the  cir- 
cumference lies  wholly  within^4BCZ>. 
Join  the  two  contours  by  a  line  ^Ll'. 
Then  ABCDAA'D'C'B'A'A  is  a 
closed  path  within  which /^  is  holo- 
morphic. 


Chap.  XVII.]  THEORY   OF   FUNCTIONS.  301 

Therefore, 

([fz .  (h  (along  ABCDAA'D'C'B'A'A)  =  0, 

or       Cfz .  dz  (along  ABC  DA)       +  Cfz .  dz  (along  .Li') 

+J}> .  dz  (along  A'D'C'B'A')  +  Cfz .  dz  (along  A'A)  =  0  ; 


but 


Cfz . dz  (along  ^.4')  =-  Cfz. dz  along  {A' A) , 

id 

Cfz .  dz  (along  A'D'C'B'A')  =  -  Cfz. dz  (along  A'B'C'D'A') 

ence 
Cfz .  dz  (along  ABCDA)       =       f/^ .  (/^  (along  A'B'C'D'A'). 


and 


Hence 


221.  That  the  integral  of  a  function  of  z  around  a  closed 
contour  enihraciug  a  point  at  which  the  function  is  infinite  is 
not  necessarily  zero  is  easily  shown  by  an  example. 

fz  = ,  t  being  a  given  constant,  is  single-valued,   con- 

z  —  t 
tinuous,  and  finite  throughout  the  whole  of  the  plane  except  at 

the  point  t,  at  which  becomes  infinite,  without,  however, 

z—t 

ceasing  to  be  single-valued. 

/dz 
around  a  circle  whose  centre   is  f,   and 
z-t 
whose  radius  is  any  arbitrarily  chosen  value  c.     If  z  is  on  the 
circumference  of  this  circle 

z  —  t  =  (.  (cos  <^  -i-  /  sin  <^) 

=  o'"  by  [.^],  An.  .-.l. 

z=-t  +  ce''"' 

and  <J-:  =  u<^-l"d<f>. 


302  INTEGRAL  CALCULUS.  [Akt.  221. 

Hence  f-^  (around  ahc)  =  T"  ^^^^  =  2  wi. 


J: 


From  what  lias  been   proved  in  Art.    220,  it  follows  that 

dz 

around  any  closed  contour  embracing  t  must  also  be 

z  —  t 
equal  to  2iTi. 

■ dz,  when  Fz  is 

z  —  t 
supposed  to  be  holoraorphic  in  the  portion  of  the  plane  con- 
sidered, and  where  the  integral  is  to  be  taken  around  any  closed 
contour  embracing  the  point  z  =  t. 

— —    is   holomorphic   except   at    the   point  z  =  t^   where    it 

becomes  infinite.  The  required  integral  is,  then,  equal  to  the 
integral  around  a  circumference  described  from  the  point  t  as 
a  centre,  with  any  given  radius  e,  that  is,  by  the  reasoning  just 

used  in  the  case  of  (  — — ,  to 
J  z  —  t 

Jo  £6*'  */» 

and  in  this  expression  e  may  be  taken  at  pleasure.     If  now  c  is 
made  infinitesimal  te**  is  infinitesimal,  and  since  Fz  is  continu- 
ous F{t  -\-  ee*')  is  equal  to  i^^  +  t;  where  ?/  is  some  infinitesimal, 
and  F{t  +  ce*')  d<^  is  equal  to  Ft .  c^ -{- -q .  d<^. 
Now,  by  I.  Art.  IGl, 

f""  {Ft .  dcf>  +  7;(?<^)  =   C'^'fi  .  d<fi. 

Hence  i  C^F{t-i-  ee*')  dcf>  =  i  i  '"Ft .  d<l>  =  2 iriFt ; 

and  we  get  the  important  result  that  | dz,  taken  around  any 

contour  including  the  point  z  =  t,  is  ecpial  to  'Im.Ft. 


From  Ihis  we  have 


1     /*  Fz 


Ft  =  J-     J^.dzi 

2  TTtJ  Z  —  t 


Cll.vr.   XVII]  THEORY   OF    FUNCTIONS.  303 

and  we  see  that  a  holomorphic  function  is  determined  every- 
ichere  inside  a  dosed  contour  if  its  value  is  given  at  every  point 
of  the  contour. 

If  in  the  formula  Ft  =    —  C^  dz  [11 

2 TriJ  z-t  *-  -• 

we  change  t  to  t  -\-  A^  we  get 

A«=J-  r>..ri/— ! L.A  =  _L  f       Fz.d..At_ 

•iTTiJ  \z  —  t-\t       Z  —  tj      '2iriJ  {z  —  t){z-t-\t) 

whence 


limi 
A<= 


0  [  A«   J       27r  J  A<  =  0  [  (^  _  <)  (2  _  «  _  AOJ 

r,,         1     C  Fz .  dz  ,  ,-, 

or  Ft  = I , ;  I  2] 

and  in  like  manner  we  get 

27rij  {z-ty  •-  ■' 

and  in  general  F^''H=  ^  ( ,,  [4] 


each  of  the  integrals  in  these  formulas  being  taken  around  a 
closed  contonr  lying  wholly  in  that  portion  of  the  plane  in  which 
Fz  is  holomorphic,  and  enclosing  the  point  z  =  t. 

Til.  The  integral  of  a  holomorphic  function  along  any  given 
path  is  finite  and  determinate,  for,  by  [3],  Art.  218,  it  is  equal 
to  the  sum  of  four  line  integrals,  each  of  which  is  finite  and 
determinate  (Art.  16G). 

If  a  series  iVq  +  «'i  +  "'2  +  "")  ^^'here  m'o,  ?/'i,  ti\  •-  are  holo- 
morphic functions  of  z,  is  uniformly  convergent  for  all  values  of 
z  in  a  certain  portion  of  the  plane,  the  integral  of  the  series 
along  any  given  path  lying  in  that  porti(m  of  the  plane  /,<  the 
series  formed  of  the  integrals  of  the  terms  of  the  given  series 
along  the  path  in  rpiestion,  and  the  neu<  series  is  convergent. 


304  INTEGRAL   CALCULUS.  [Aut.  223. 

For,  let      S  =  W(i -\- Wi -{-  W2 -\ \-  "'„  +  "'„  + 1  -| — 

=  Wo  +  ^^1  +  «'2  H h  ">.  +  ^'«) 

where  E„  =  iv„  +  i-\-  w„  +  2+-, 

and  where  by  hypothesis  71  may  be  taken  so  great  that  the 
modulus  of  R^  is  less  than  c  for  all  values  of  z  in  the  portion 
of  the  plane  in  question,  e  being  a  positive  quantity  taken  in 
advance  and  as  small  as  we  please. 

CSclz  =  Cwo  dz  +  Cwi  dz  H 1-  fw^,,  dz  +  C R„  dz 

for  any  given  value  of  n. 

By  the  proposition  at  the  beginning  of  this  article.  |  Sdz 

along  the  given  path  is  finite  and  determinate,  as  are  also 

I  w'o  dz,   I  n\  dz,  etc. 

The  modulus  of  |  R^dz  is  not  greater  than  the  line-integral 

along  tlie  given  path  of  the  modulus  of  R^  (v.  Art.  218).  If, 
now,  n  is  taken  sufficiently  great,  each  value  of  the  modulus 
of  i?„  will  be  less  than  e;  consequently  each  element  of  the 
cylindrical    surface    representing    the    line-integral    of    the 

modulus  of  i2„  will  be  less  than  e  (v.  Art.  166),  and  |  R,^dz 

will  be  less  in  absolute  value  than  c  multiplied  by  the  length 
of  the  path  along  wliich  the  integral  is  taken. 

Therefore,    C Sdz  =  Cwo  dz  +  Civi  dz  +  Ciiu  dz-\--', 

and,  since  the  first  member  is  finite  and  determinate,  the 
second  member  is  a  convergent  series. 


Taylor^ s  and  3farl<nirin\<i   Theorems. 
223.  ^^  =  1  +  y  +  ^2  +  ,/  -f  ...  y»-» 

identically,  if  "  is  a  positive  integer,  even  when  q  is  imaginary 


CiiAP.  XVII.] 


THEORY    OF    FUNCTIONS. 


30.") 


If  the  iikhIuIus  of  q  is  less  than  1, 
limit 


Hence 


1  1       "    1       •?   ,  \'\\n\i  V\  —  f/"!  1  rn 

+''+''"+''  +  ■■■  =  »= 4 i-';J=r=^-  ^'- 

even  when  q  is  iniaginnrv,  provided  that  the  modulus  of  q  is 
less  than  1. 

Suppose,  now,  that  everj'where  within  and  on  a  certain  cir- 
cumference   described    with    the  point  z  =  a  as  a  centre  Fz  is 
holomorphic.     Let  z=  i  be  any  point  within  this  circumference, 
and  z  =  Z  be  a  point  on  the  circum- 
ference.    Then  the  modulus  of  Z  —  a 
is  the  distance  from  a  to  Z,  and  the 
modulus  of  t  —  a  is  the  distance  from 
a  to  ^ ; 
hence     mod  {t  —  a)  <  mod  {Z  —  <i), 

and         modf  — — —  )<  1- 


Z  —  t      Z  —  a  —  {t  —  a)      Z 


Z  —  a 


Hence 

1     ^      1 
Z-t      Z- 


Z-a[_    ^Z-rt       (Z -«)•-'       (Z- a)-'  J" 

I'V  [1] 


^',,  +  4^«)>il=1^4--..  [2] 


-a      (Z-a)-      {Z-af      {Z  -  a)* 
and  the  second  member  of  [2]  is  a  convergent  series. 

Multiply  [2]  bv  ^,  and  the  series  will  still  bo  convergent 
for  each  value  of  z  which  Ave  have  to  consider ;   we  get 


1       FZ 
•2  Tri    Z  -  t 


27r/[_Z—  a 


+  {f 


FZ 


'  (Z-a)-      ^  (Z-a)' 


[••'] 


306  INTE(;UAL   CALCl'LUS.  [Aht.  223. 

Integrate  now  botli   nu'inbers  of  [H]   uroiiud  the  circumfer- 
ence, and  we  have 


J. 
27ri 


iJ  Z-t  •'■^i[J  Z-a  ^  'J  (Z-h)' 

+  <'-">'/(^'"^+-]^  w 

and,  since  each  of  the  functions  to  be  integrated  is  holoniorphic 
on  the  contour  around  which  tlie  integral  is  taken,  and  tlie 
second  member  of  [3]  is  convergent,  each  integral  will  be  Unite 
and  determinate,  and  the  second  member  of  [4]  will  be  con- 
vei-gent. 

Substituting  in  [4]  the  values  obtained  in  Art.  221,  [1],  [2], 
[3],  and  [4],  we  have 

Ft  =  Fa  +  {t-  a)  Fa  +  i!-Z^F'a  +  itr_l^F"'a  +  .-. 

If  the  point  z  =  a  is  at  the  origin,  a  =  0  and  [5]  becomes 
Ft=Fo  +  tFo  +  il  F'n  +  ^^  F"o  +  ...,  [C] 

which  is  Maclauriu's  Theorem. 

That  [5]  is  merely  a  new  form  of  Taylor's  Theorem  is  easily 
Been  if  we  let  t  —  a  =  h,  whence  t  =  a  -\-h.  and  [a]  becomes 

F{a  +  h)  =Fa  +  h  F'a  +  ^f^F"a+^F"'a  +  ....  [7] 

[fl]  can,  of  course,  be  written 

Fz  =  Fo  +  zF'o  +  |1  F"o  +~i'""o  +  -,  [8] 

and  [/i]  as 

Fz  =  Fi  +  (z  -  a)  F'a  +  ^~  ~  'p^F"a  +  i^^p^V"'a  +  •  •  • ; 


Chap.  XVII.]  THEORY   OF    FUNCTIONS.  3()T 

aud  we  get  the  veiT  important  result  that  if  a  function  of  z  /.•! 
holomorphic  ivithin  a  circle  n-hose  centre  is  at  the  oriyin  it  may 
he  developed  by  Maclaurin's  Theorem^  and  the  development  will 
Jiold,  that  is,  the  series  will  be  convergent,  for  all  vabtes  of  z 
lying  icithin  the  circle. 

If  a  function  of  z  is  holomorphic  within  a  circle  described 
from  z  =a  as  a  centre  it  can  be  developed  by  Taylor's  Theorem 
into  a  series  arranged  according  to  powers  of  z  —  a,  aud  the 
development  will  hold  for  all  values  of  z  lying  within  the  circle. 

The  question  of  the  convergeucy  of  either  Taylor's  or 
Maclaurin's  Series  for  the  case  when  z  lies  on  the  circum- 
ference of  the  circle  needs  special  investigation,  and  will  not 
be  considered  here. 

If  the  function  which  we  wish  to  develop  is  single- valued,  in 
drawing  our  circle  of  convei'gence  we  need  avoid  only  those 
points  at  which  the  function  becomes  infinite ;  but  if  it  is 
multiple-valued  we  must  avoid  also  those  at  which  its  derivative 
is  zero  or  infinite  (v.  Art.  217). 

224.  "We  are  now  able  to  investigate  from  a  new  point  of 
view  the  question  of  the  convergence  of  the  series  obtained  by 
Taylor's  and  Maclaurin's  Theorems  in  I.  Chap.  IX. 

Let  us  begin  with  the  Binomial  Theorem, 

(a)    (a-f//)"  =  a"  +  7(a»  '/t -fn  ^"  ~  ^  ^r-'/r  +  •-,  [1] 

or,  following  the  notation  of  [!»],  Art.  220, 

z'^  =  a"-\-mr-\z-a)-\-''^'\-^\,"-\z-ar+....     [2] 

If  n  is  a  positive  integer,  2"  is  holomorphic  throughout  the 
whole  plane,  and  [2]  holtls  lor  all  values  of  z  and  a,  and  [1] 
for  all  values  of  a  and  h. 

If  n  is  a  negative  integer,  2"  is  single-valueil,  and  it  i.s  liuito 
and  continuous  except  for  2  =  0,  where  2"  becomes  infinite. 
[2]  is.  then,  convergent  for  all  values  of  z  lying  within  a  circle 
described  with  a  as  a  centre  and  passing  through  the  origin  ; 


308  INTEGRAL   CAL(^ULUS.  [Art.  224. 

tbat  is,  for  all  values  of  z,  such  that  mod  {z  —  a)  <  mod  a  ;  and 
consequently  [1]  holds  if  viodh  <.moda. 

If  71  is  a  fraction,  z"  is  multiple- valued,   and   our  circle  of 

convergence  must  avoid  the  points  at  which  —  becomes  zero 

dz 
or  infinite  ;  but  as  the  origin  is  the  only  point  of  this  character, 
the  circle  of  convergence  is  the  same  as  in  the  case  last  con- 
sidered, and  [1]  holds  for  all  cases  where  modh<modn. 

When  a  and  h  are  real  our  results  agree  with  those  obtained 
in  I.  Art.  131. 

(^h)  €'  =  €'+>'' =  e' (cosy +  is\nij)  ([4J,  Art.  31) 

is  single-valued  and  continuous,  and  becomes  infinite  only  when 
x=(x.  Maclauriu's  development  for  e'  holds,  then,  for  all 
finite  values  of  z. 

(c)  logz  =  log  (r cis^)  =  logr  +  <l>i  (Art.  33) 

is  finite  and  continuous  throughout  the  whole  plane.      It  is, 

however,  multiple-valued,  but  its  derivative  -  becomes  infinite 

z 
only  when  z  =  0,  and  does  not  become  zero  for  any  finite  value 
of  z.  logz,  then,  can  be  developed  into  a  convergent  series, 
arranged  according  to  powers  of  z  —  a,  for  all  values  of  z  within 
a  circle  having  the  centre  a  and  passing  through  the  origin  ; 
that  is,  for  all  cases  where  mod  (z  —  a)<  mod  a. 
If  z  —  a=h,  we  get 

log(a  +  /0  =  loga-h^-^,  +  ;^,--^^-h...,  [3] 

[3]  holding  for  all  cases  where  modh  <.moda. 
If  a  =  1  and  h  =  z,  we  get 

log(l +z)  =  ^-- -[----  +  •••,  [4] 

wliich  liolds  for  all  values  of  z  where  mod  z  <C  1. 

(d)  sin  z  =  sin  {x  -\-  yi)  =  sin  x  .  — ' 1-  /  cos  x  • , 


CiiAi.  XVII.]  Til EOllY    OF   FUNCTIONS.  300 

ami       eos2  =  cos  {x  +  y/)  —  cosx  •  — ^ ;"  isin  x  •  — , 

(v.  [3]  ami  [4],  Art.  ;$.-)) 
are  single-valued,  ami  arc  finite  aud  continuous  throughout  the 
plane.  Therefore,  JMaelauriu's  developmeuts  for  sin  2;  aud  cos 2 
hold  for  all  values  of  z. 

(e)    tan2;  = ,    ami   sec;;;  = ,    are    single-valued    aud 

cos  2  cos  2; 

continuous,   aud  become  infinite  onW  when  cosz  =  0;  that  is, 

when  2  =  ^.     Therefore,  Maclauriu's  developments  for  tan 2;  and 

sec^  (I.  Art.  138),  hold  for  every  value  of  z  whose  modulus  is 

less  thau  ^^ 
2 

(/')    ctnz= ,  and  csc2  = become  infinite  when  2  =  0, 

sinz  s'lnz 

and  cannot  be  developed  by  Maclauriu's  Theorem. 

(g)    sin  'z  is  finite  and  continuous  throughout  the  plane;  it 
is,  however,  multiple- valued,  and  its  derivative -y=^  becomes 

infinite  when  z  =  'l,  and  wlien  z  =  —  1 .  Therefore,  the  develop- 
ment for  sin~^2  (I.  Art.  135  [2]),  holds  for  any  value  of  z 
whose  modulus  is  less  than  1. 

(h)    tan"^2;  is  finite  and  continuous  throughout  tiie  plane  ;  it 
is  multiple-valued,   and   its  derivative becomes   infinite 

when  z=  i,  and  when  z  =  —  i.  Therefore,  the  development  for 
tan  '2  (I.  Art.  135  [1]),  holds  if  mod z<  modi;  that  is,  if 
modz  <  1. 

EXAMI'I.KS. 

(1)  Show  that  the  development  of f-  log  ( 1  +  z),  given 

in  I.  Art.  136,  Ex.  1,  holds  if  nuxU<  1. 

(2)  Show  that  the  development  of  l<>g  ( 1  -f-'''),  given   in   I. 
Art.  136,  Ex.  2,  holds  if  mod2;<7r. 

(3)  Obtain  the  following   developim-nts,   mikI   fin.l   f-.r  what 
real  values  of  x  they  hold  good  : 


INTEGRAL   CALCULUS. 


[Art.  224. 


#.    ,        ,      ,       /  .,   ,      .,.        ,  ,    X       1        x^     ,    1.3        X^ 


^f^    e'.coso; 


-^"(l+y;+»(3«-2)^-,+ 


=  1  + 


2x^      Ax* 
3!        4!  ' 


^ 


_31^     \ 
6!   •••/ 


\/)  tannic 
(j/)    (1  +  2x  +  3.-^2)72  =  1  _  X-  +  2.1'"' 


/,       x~  ,  4  a;*      31  a/* 

V        2!  4! 

,    .r-    ,  2;tr"'  ,   9x^ 

3  5 


U()    log 


l-.'r; 


, , ,      ,  .^"^  ,  ar' 
'  3       5 


//   .  ,    2    ,  ,    2'    ,  ,   2M7   ,  ,  2«.31    ,  ,  2^691    „ 

I  ^)   tiinx=x-\ x^-] x-'-l .»•'  H x/-\ .t"  ... 

\/^^  3!         5!  7!  9!        ^     U! 


(A-)    X .  etn  X  =  \  —  '- 


2.T« 


2a-^« 


3      4.")      iilf)      4725      93555 


(0 


log  tanf    +a;  )  =  lo«r  tan    +2.''+     x^-i x'-\ x'  ••• 

\4       J        ^         1  3!         5!  7! 


/     N     ,i„x       1    ,    ^    ,    •*■"       3.<;*      8.*^      3a-''  ,   5()a-'  , 
^    ^  1  !      2 !       4  !        5 !        6 !         7 ! 

^  2!3!4! 


5! 


77  a.-^ 
6! 


(o)    (ver.in-.r  =2f.r  +  i^  +  ^-^  a^      1^^  a^  N 

^   ^     ^  '^  V        3.2      3.5   3       3.5.7   'i  J 


2   .  2a-  ,  X-      2. 


2—2  .>:  -|-  ;r      4        4        4 


•J-         4-        4- 


^^^    G-5.r+.x-='      2-.T      3 -a;      ^^2      ^^J      V2''      S^J 


riiAi'.  XVII.]  THEORY   OF    FUNCTIONS.  31 1 

Aiisicers. 
{(()    —a<x<a; 

^l)    -  X  <  X  <  X  if  /t  >  0,   -  ^  <  X  <  ^  if  7i  <  0  ; 
(o)    -oc<x<x;  ((/)    -x<x<x; 

(e)    -1<.T<1;  (/)   -^<.r<;;; 


(0     -l<.f<l;  (./)    -:^<x< 


(k)    -7r<.T<7r;  (0     -"<a;<"; 

■1  4 

(m)  —  X  <  X-  <  00  ;  ('0    —  ^  <  •^"  <  J  ; 

(o)    -2<.f<2;  0>)   -V2<.i-<V2 

(g)    -2<a;<2. 


312  INTKGUAL    CALCULUS.  [Akt.  22&. 


CHAPTER   XVIII. 

KEY  TO   THE   SOLUTION    OF   DIFFERENTIAL    EQUATIONS. 

225.  In  this  chapter  an  analytical  ke}'  leads  to  a  set  of  con- 
cise, practical  rules,  embodying  most  of  the  ordinary  methods 
employed  in  solving  differential  equations  ;  and  the  attempt  has 
been  made  to  render  these  rules  so  explicit  that  they  may  be 
understood  and  ai)plied  by  any  one  who  has  mastered  the  Inte- 
gral Calculus  proper. 

The  key  is  based  npon  "Boole's  Differential  Equations" 
(London  :  Macmillan  &  Co.),  to  which  or  to  "  Forsyth's  Differ- 
ential Equations"  (London:  Macmillan  &  Co.),  we  refer  the 
student  who  wishes  to  become  familiar  with  the  theoretical 
considerations  upon  which  the  working  rules  are  based. 

226.  A  differential  equation  is  an  expressed  relation  involv- 
ing derivatives  with  or  without  the  primitive  variables  from 
which  they  are  derived. 

For  example  : 

(H-.T)2/-f(l-2/)xg  =  0,  (1) 

x'M_a!,  =  x  +  l,  (2) 

ax 


^_y  +  a-V.r-'-r  =  o,  (8) 


dx 


(f5"=4+^:!9-  <^' 


^  _  2'J^  -f  2 ''^  -  2 ''•'  +  .V  =  1 , 
rix*         d.r^         dx-         dx 


(^) 


Cii.vr.  XVIIl.j  DIFFERENTIAL   EQUATIONS.       KFY.  318 

sin^x— f +  sin.rcos.^-  •'  —  >/  =  x  —  ii'ii\x,        (0) 
dx-^  dx     '  ^ 

x{\-xyj^- -2  ;/  =  (),  (7) 

B/z  -  crDJ^z  =  U,  (9) 

are  differential  equations. 

Tlie  order  of  a  differential  equation  is  the  same  as  that  of  the 
derivative  of  highest  order  which  appears  in  the  equation. 

Equations  (1),  (2),  (3),  and  (4)  are  of  the  first  order ;  (6), 
(7),  (8),  and  (9)  of  the  second  order;  and  (o)  of  the  fourth 
order. 

The  degree  of  a  differential  equation  is  the  same  as  the  power 
to  which  the  derivative  of  highest  order  in  the  equation  is 
raised,  that  derivative  being  supposed  to  enter  into  the  equation 
in  a  rational  form. 

p:quations  (1),  (2),  (3),  (5),  (6),  (7),  (S),  and  (9)  are  all 
of  the  first  degree  ;   (4)  is  of  the  third  degree. 

A  differential  equation  is  linear  when  it  would  be  of  the  first 
degree  if  the  dependent  variable  and  all  its  derivatives  were 
regarded  as  unknown  quantities. 

Equations  (2),  (5),  (G),  (7),  (8),  and  (9)  are  linear. 

The  equation  not  containing  differentials  or  derivatives,  and 
expressing  the  most  general  relation  between  the  primitive  vari- 
ables consistent  with  the  given  differential  equation,  is  called 
its  general  sohdion  or  complete  prlmitioe.  A  general  solution 
will  always  contain  arbitrary  constants  or  arbitrary  functions. 

The  differential  equation  is  formed  from  the  complete  primi- 
tive by  direct  differentiation,  or  by  differentiation  and  the 
subsequent  elimination  of  constants  or  functions  Itetween  the 
primitive  and  tiie  derived  (-(luations. 

If  it  has  been  formed  by  ditferentiation  onh/  without  sub- 
sequent elimination  or  reduction,  the  differential  ecpiation  is 
said  to  be  exact. 


314  INTEGRAL  CALCULUS.         [Art.  227. 

A  suigular  solution  of  a  differential  equation  is  a  relation 
between  the  primitive  variables  which  satisfies  the  differential 
equation  by  means  of  the  values  whicli  it  gives  to  tlie  deriva- 
tives, but  which  cannot  be  obtained  from  the  complete  primitive 
by  giving  particular  values  to  the  arbitrary  constants. 

227.  We  shall  illustrate  the  use  of  the  key  by  solving  equa- 
tions (1),  (2),  (3),  (4),  (5),  (6),  (7),  (8),  and  (9)  of  Art.  226 
by  its  aid. 

(1)  (l+x).v-f(l-y).x-^=0,  or  {\+x)ydx+{\-y)xdy=Q. 

clx 

Beginning  at  the  beginning  of  the  key,  we  see  that  we  have  a 
single  equation,  and  hence  look  under  I.,  p.  326 ;  it  involves 
ordinary  derivatives :  we  are  then  directed  to  II.,  p.  326 ;  it 
contains  two  variables :  we  go  to  III.,  p.  326 ;  it  is  of  the 
first  order,  IV.,  p.  326,  and  of  the  first  degree,  V.,  p.  326. 

It  is  reducible  to  the  form 

--!—  dx  -i ■-  dy  =  0, 

X  y 

which  comes  under    Xdx  +  Ydy  =  0. 

Hence  we  turn  to  (1),  p.  330,  and  there  find  the  specific  direc- 
tions for  its  solution.     Integrating  each  term  separately,  we  get 

log  x-\-x->r  log  ?/  —  y  =  c,     or     log  (xy)  -{-  x  —  y  =  Cy 

the  required  primitive  equation. 

(2)  x^^^-ay  =  x-\-l. 

Beginning  again  at  the  beginning  of  the  key,  we  are  directed 
through  I.,  II.,  HI.,  IV.,  to  V.,  p.  326.  Looking  under  V., 
we  see  that  it  will  come  under  either  the  third  or  tlie  fourth 
head.  Let  us  try  the  fourth;  wo  are  referred  to  (4),  p.  3.S0, 
for  specific  directions. 

Obeying  instructions,  the  work  is  as  follows : 

dx 


CUAr.  XVIII.]         DIFFERENTIAL   EQUATIONS.      KEY.  .^15 

xdii  —  ayiLr  =0, 
d^i  _  oiix    _  , 
y        ^^' 
log?/  —  a  log.v  =  c, 

log^'^  =  c; 
x" 

a;" 

y    =C.t-,  (1) 

rfv       /<  .  1  ,    ..(10 


dx  d.i 

Substitute  in  the  given  equation, 

aCx"  +  x''+'^^-  aCx"  =  .r  -f- 1 , 
dx- 

a;»+i— -(.r+l)  =  0, 
da; 

dC--^'+/f/a;  =  0, 

(a  —  \)x"  '      o.r ' 
Substitute  this  vaUie  for  C  in  (1),  and  we  get 

\a      a— I J 
the  n'(iuired  primitive. 

(3)  x^  -  V  +  A-  V.r=-*  -  V-  =  0. 

(/or      ■ 

Beginning  at  the  l)egiiuiing  of  the  key,  we  are  directed 
through  I.,  II.,  III.,  IV.,  to  v.,  pjigo  32G.  Looking  under  V., 
we  find  tlmt  our  equation  does  not  come  under  any  of  the 
special  forms  tlu-re  given.  We  are  consequently  driven  to 
obtaining  a  sohition  in  the  form  of  a  series,  and  for  specific 
instructions  we  are  referred  to  (13),  page  332.  Obeying 
these,  our  process  is  the  following : 


316  INTEGRAL  CALCULUS.  [Art.  22?. 

#     _.V  ,/72 -2  dyo     _  .Vn  _  W      2  _  -.  2 

da;       a:  dxo       .t„ 

-7-;^  —  —  y V.X    — »/,  -7—7  —  —  Z/o V  .Tf,    —  T/o  , 


cLc^  X  dxo  3*0 

da;'  a;  da^o  a'o 

and  the  general  value  of  ?/  is 

y  =  yo  +  {x-  X,)  (^1  _  V.V3^^)  -  ^^^^'(z/o  +  |/.V^^^) 

(a^  -  a;n)'' /3  ?/o       /-^ -A 

(a;-a;o)*  /         4    /^ ,\      (x-a'n)^  /5  j/o        /-;; , 

+  -^T-[^'^  +  ;^  ^^o-yo-J  +  ^T"(^ir  -  Vxv  -  yo 


This  result  can  be  very  greatly  simplified  by  breaking  up  the 
series  ;  we  have 

-^      ^\  2!  4!  G! 

( x-a-„)  /        (X  -  a;,,) '^  ,   (a;  -  a;,,) *      (.f  -  x^^ 

^•'""^T^l       2!    +'"~4!         (rr~ 

-^   V  3!  5!  7!  / 


CiiAP.  XVIII.]         DIFFERENTIAL   EQUATIONS.      KEY.  31 


-  ViV 


^•(^")(^^-'"'-'^^*''5T^' 


7! 


ft 


V^ 


COS  (x  —  a^o) — si'i  {'C  —  a",i) 


m' 


'] 


=  a;M^  cos  (.r  —  a^o)  —  Jl  —  ^  •  sin  (x-  —  a-o)   . 
|_a*o  ^        ajy  J 

^  is  entirely  arbitrary  ;  call  it  sin  a,  then 

Xo 

y  =  .r[siua  cos(a;  —  x^)  — cosa  sin(.r  —  .^\|)]  =  xsin[a— (a;  — aV)], 
y  =  X  sin  (o  —  a;) ,  whore  c  is  any  constant. 


(4) 


^'=4-^- 


Beginning  at  the  beginning  of  the  key,  we  are  directed 
throngh  I.,  II.,  III.,  IV.,  to  VI.,  page  327.  Looking  under  VI. 
we  see  that  the  equation  is  of  the  first  degree  in  x ;  we  are 
referred  to  (17),  page  334,  for  our  specific  instructions. 

Obeying  these,  we  first  replace  —  by  p  ;  the  equation  becomes 
"  dx    ' 

p'^  =  y*  {y  +  •''"p)  • 

Differentiate  relatiyely  to  ?/.  and  we  get 

^j/h'  =  4^  (y  +  a:^0  +  ^!/'  +  -^V'  v* 
dy  dy 


Eliminate  a;, 


dy        y  \         I'Jdy 

p      (fy  y 


318  LNTEG UAL   CALCULUS.  [Akt.  227. 

Striking  out  the  factor  2y.'  +  y"',  we  have 
1^-^  =  0, 

P  dy     y 

a  differential  equation  of  the  first  order  and  degree  in  which 
the  variables  :iic  separated,  and  which  therefore  can  be  solved 
by  (1),  page  314. 

Its  solution  is  log\/>  —  log//-  =  C, 

P 
or  —  =  c. 

y- 

Kliniinating  p  between  this  and  the  given  equation,  and  re- 
ducing, we  have  c?/ (.r  —  c'-)  =  1 ,  as  our  required  solution. 

(5)  'l|_2^  +  2f?-2f'  +  y=l.  (1) 

ax*        clx^        dx-         dx 

Beginning  at  the  beginning  of  the  key,  we  are  directed 
through  I.,  II.,  III.,  VII.,  to  (22)  (a),'  page  335,  for  our 
specific  directions. 

We  see  at  once  that  y=l  is  a  particular  solution. 

Obeying  directions,  we  have  now  to  solve 

Let  y  =  e""",  and  we  have 

m*  -  2 ?«•■'  -f  2 m"^  -2m-\-\=0, 
as  our  auxiliary  algebraic  equation  in  m.     Its  roots  are 

1,  1,  V^^,  -V-H". 

The  solution  of  (2)  is  then 

l(  =  (A  +  Bx)  c'  4-  C cos x-\-  D  sin .r, 
and  of  (1)  is 

2/  =  ( J  +  Bx)  c'  +  C cos ..•  -f-  D  sin  x  +  1. 


Chap.  XVIU.]  DIFFEKENTIAL   EgUATiuNS.      KKV.  3U* 

(6)  siu-.f  ,  •„  +  siiLi;  COS X  ,' —  if  =  x  —  sin  x.  (1) 

(Lr  ax 

Beginning  at  the  beginning  of  the  key,  we  are  directed 
through  I.,  II.,  III.,  VII.,  to  (24),  page  337,  for  onr  specific 
instructions. 

Dividing  through  by  siu-.r,  the  equation  becomes 

cPy  ,     ^      (/y  ,  ,  ,n\ 

-r^  +  ctn  x-^ —  CSC-  X  .y  =  x  esc-  x  —  esc  x.  (2) 

dxr  clx 

y  =  etna;  is  found  by  inspection  to  be  a  solution  of 

dry        ^      (hi  ,  „ 

--4  +  ctn  x~  —  csc-.r  .  w  =  0  ; 
tZjf  dx 

(2)  can  then  be  solved  by  (24)  (a). 

Substitute  y  =  2;ctn.i-  in  (2),  and  it  becomes 

ctu  x —  +  (ctn- a;  —  2  csc^x)  —  =  x  csc-.r  —  esc  a;, 
rfar  dx 

or         — ■  —  (tau.i;  + secx'csca;) —  =  a;  sec  x  esc  a;  — seer.       (3) 
dir  dx 

Referring  to   (25),  page  339,  and  obeying  instructions,  we 

dz 
let  z' =  — ,  and  (3)  becomes 
dx 

dz' 

(tan.r  +  sec  a;  esc  a*)  z'  =  x  seer  cscr  —  seer, 

dx 

a  linear  differential  equation  of  the  first  order  in  z'.  whose  solu- 
tion by  (4),  page  330.  is 

z'  =  A  tanr  seer  —  r  sec^r  +  tan  .r  secf  (log  tan" —  log  sin.r)  ; 

but  z'  =  — ,  whence  integratiiiii^,  we  have 
dx 

z  =B-{-  Asecx  —  xtiinx  —  (1  -f  seer)  log  (1  +  cos.r), 

and 

y  =  A  esc  r  +  J5  ctn  x  —  x  —  (esc  .»•  +  ctn  x )  1< )g  ( 1  -|-  cos  x) . 


320  INTEGRAL  CALCULUS.  [Akt.  227. 

(7)  .^i-x)-pl-2y  =  0. 

d.i" 

Beginning  at  the  beginning  of  the  key,  we  are  directed 
through  I.,  II.,  III.,  VII.,  to  (24),  page  337,  for  our  specilic 
instructions. 

Let  us  try  the  method  of  (24)  (e),  page  339. 

Assume  y  =  '2,a„x"\  and  substitute  in  the  given  equation; 
we  have 

2 [m  (m  —  1 )  a,„ af  ^  —  2m  {m  —  1 )  a^cc"* 

+  ?/i  (m  —  1 )  a„,  0?"*+^  —  2  a,„  a;"*]  =  0. 

Writing  the  coefficient  of  ic""  in  this  sum  equal  to  zero,  we 
have 

m  (m  +  1)  a„.+,  -  2[m(m-l)  +l]a„  +  (m  -  l)(m  -  2)  o,„_i  =  0, 

and  we  wish  to  choose  the  simplest  set  of  vahies  that  will 
satisfy  this  relation. 

Substituting  m  =  0,  ?h  =  —  1,  m  =  —  2,  etc.,  in  this  relation, 
we  find 

a_i^c'o.  a_2=a_i,  tf_3  =  (x_2,  •••. 

Hence  if  we  take  Oo=  ^i  it  follows  that 

a_  1  =  a_2  =  rt_:5  •  •  •  =  0, 

and    no    negative    powers   of   x   will    occur   in   our   particular 
solution. 

Substituting  now  ?)i  =  1,  ?/i  =  2,  m  =  3,  etc.,  we  have 

tti  =  ttg  =  «3  =  O4  =  •  •  • . 

Taking  aj  =  1,  we  get  as  our  required  particular  solution  of  the 
given  equation 

y  =x  +  ar  +  .r'  +  ic*  +  •••. 

This  can  be  written  in  finite  form,  since  we  know  that 

1  +.r  +  .x-2  +  ar''...  =  — ? — 
1—x 

Hence  y  = 

^      l-x 

is  a  pailicular  solution. 


CiiAP.  XVTII.J  DIFFERENTIAL   EQUATIONS.      KEY.  321 

Turning  now  to  (24)  (a),  page  337,  we  fiml 

y  =  r-^^  +  C-'  (  1  +  X  +  — -—  log.x*  ]. 
I  —X  \  i  —  X 

Beginning   at   the   beginning   of   the   key,  we   are   directed 
tln-ough  I.,  II.,  III.,  VII.,  to  (24),  page  337,  for  our  specific 
instructions.     Let  us  trv  again  the  method  (24)  (e),  page  339. 
Assume  2/  =  Sa^x"",  and  substitute  in  the  given  equation, 
2[??i  {m  —  l)a„.af --  +  a^.-c'"  —  2  a„.x™  ^j  =  o. 
The  terms  containing  af  are 

(??i  +  2)  {m  +  1 )  a^+'iX'^  +  a,^x'^  —  'ia^.^^^ ; 
writing  the  sum  of  the  coefficients  equal  to  zero,  we  have 

m(m  +  3)a„,^2  +  «».  =  0-  (1) 

Letting  m  —  0  and  m  =  —  3,  we  get  Oo  =  0  «'^ud  a  3  =  0;  and  all 
terms  of  y  involving  even  negative  powers  of  x  disappear,  as  do 
all  terms  involving  odd  negative  powers,  except  the  —  1st. 

In  general  «„-.o= (2) 

m  {ill  +  3) 


From  this  we  get 

a2_ 1_ 

"'  ~      2^5  ~      3 !  0 


1 


•J!  11 

Hence  y  =  -^--^     +  -—  -  -^~- 

3      3  !  o      5  !  /       /  !  i) 


if  we  take  a., 


a.. 

2.4. .0.7 

«2 

2.4.5.6.7 

.9 

flj 

2.4.5.0.7 

.8.1) 

.11 

7    -^- 

X* 

+  - 

is  a  particular   solution  of  the   given  equation.     This  can  be 
tiirown  into  liuile  form  witlioiit  nuicli  hibor. 


J2 

INTEGIl 

\L 

CALCULUS. 

We  have 

xy 

_X^ 

3  ' 

X' 

3!5 

+4 

0  I 

7      7!9 

d{xy) 
dx 

=  x2_ 

x' 
~3! 

+ 

5! 

x^      x-'o 
7!"^9! 

[Akt.  227. 


9!  11 


/        x"   ,   X?       x'x^\ 


=  .rsina; ; 
whence  xy  =  sin. x-  —a;  cos  re, 

and  y  =  -(sina;  — iccosa;). 

X 

Bj' going  back  to  (2),  and  using  odd  vahies  of  m,  we  get 
another  solution  of  our  given  equation,  namely, 

—  i_L^__£iL        •^-"'          x^ 
^~  X      2      2!44!G      61S' 

which  can  be  reduced  to 

y  =  -  (cos  X  +  X  sui  X) . 
re 

Hence  our  complete  solution  is 

y  =  -[A{cosx-\-  ajsin.r)  4--S(sin.r  — rccosa;)], 

X 

y=AS2lll=^  +  sin  (x-cA 
if  we  let  -4  =  tauc. 


(9)  DJ'z-a-DJ'z  =  0. 

Uoginning  at  the  beginning  of  the  key,  we  are  directed 
through  1.  and  IX.  to  (45),  p.  347,  for  our  specilic  iustruc- 
tions. 


Chap.  XVIII.]         DIFFERENTIAL    i:(HATl(tN8.      KKV.  323 

Obeying  these,  our  work  is  iis  follows : 
cly-  —  a-d.ir  =  0, 

chj  -  adx    =  0,  (1) 

dy  +  adx    =  0,  (2) 

dj)dy-a-dqdx  =  0.  (3) 

Combining  (1)  ami  (3),  we  get 

dpdy  —  adqdy  =  0, 
or  dp  —  ad(j  =  0.  (4) 

(1)  gives  y  —  ax  =  a. 
(4)  gives                           p  —  oq  =  ^. 

(2)  and  (3)  give  us,  in  the  same  way, 

y-\-ax  =  ai, 
p  +  aq=(3i; 
and  our  two  first  integrals  are 

p-aq=.t\{y-ax),  (5) 

P  +  f(q=f,{y  +  ax),  (6) 

/",  and  f,  denoting  arbitrary  functions. 
Determining  p  and  7,  from  (o)  and  (G), 

P  =  ^  1/2  {y  +  ox)  +/,  {y  -  ax) ] , 

g  =  -L  [/,  {y  +  ax)  -/,  {y  -  ax)]  ; 
'2  a 

dz  =itlMy^-(fx)  +j\{y-a.,^-]  ^'•'- +./^^  [./;.(.'/+"-*^')-/i(.y-«^)]^3/ 

/,  {y  +  ax)  ((/.y  +  adx)  —J\  (//  -  ax)  {dy  -  adx) 
~  '  2a 

Hence,  z  =  F{y  +  ax)  +  F,  (//  -  ax) , 

where  F  and  F,  denote  arbitrary  functions  obtained  by  integral 
ing/i  and/,,  which  are  arbitrary. 


324  INTKGUAL   CALCULUS.  [Akt.  228. 

228.  When  a  tliffeiential  equation  does  not  come  under  any 
of  the  forms  giveu  in  the  key,  a  change  of  dependent  or  inde- 
pendent variable,  or  of  both,  will  often  reduce  it  to  one  of  the 
standard  forms.  No  general  rule  can  be  laid  down  for  such  a 
substitution.  It  will,  however,  often  suffice  to  introduce  a  new 
letter  for  the  sum,  or  the  difference,  or  the  product,  or  the 
quotient  of  the  variables,  or  for  a  power  of  one  or  of  both. 
Sometimes  an  ingenious  trigonometric  substitution  is  effective, 
or  a  change  from  rectangular  to  polar  coordinates  ;  that  is,  the 
introduction  of  rcos  cf>  for  x  and  rsin^  for  y. 

The  following  examples  of  such  substitutions  are  instructive. 

(A.)      Ch((vge  of  dei^endent  variable. 

(1 )  (x -f- 1/)-'-^  —  ((-,  reduces  to  — /h  —  dx  —  U, 

dx  a-  +  z- 

if  we  introduce  z  =  x  +  //. 

(2)  —  =  sin(c/)  -  0) ,  reduces  to — d<t>  =  0, 

c/<^  1  —  sin  w 

if  w=tf>-6. 

(;'))  {x  —  y-)dx  +  2 xifdii  =  0,  reduces  to  {x  —  z)  dx  +  xdz  =0, 
\i  z=  ?/-. 

(4)    x^ly  -y  +  x  V.ir'  -  y'  =  0,  reduces  to       ^^^      +  dx  =  0, 
dx  •  Vl  -  z- 

ifz  =  y. 


(•^^) 

d-tf 

dx- 

'2  dy 
+  -  -f-n-ii  = 
x  dx 

0,  reduces 

dx- 

-n'z  = 

0, 

if 

z  = 

Xl/. 

(B.)      ClKwge 

of  indej)e) 

dent  variable. 

(1) 

(1- 

-.Tg  +  ,= 

0,  reduces 

to 

eos-^^  +  sin^c 

-^>^ 

=  0,  if 

X  =  sin 

8. 

CiiAP.  XVIII. j       diffp:rential  equations,     key.        32c 

(2)  ^^'>'  +  tau x^  +  COS-.K .  ^  =  0,  reduces  to  '-^  +  y  =  0, 
cIt-  dx  ilz- 

if  2  =  sin  X. 

(C.)      CIi(iur/e  of  both  variables. 

to 


(1)    (\-  '^.rn  =  ^  (.r  -  y-"  -  (<■-') ,  reduce. 
^^       dx-J         ax 


V  —  z—  —  —(z  —  V  —  (('-')  =0,   if  z  =  .r-  and  v  =  y- 
dz-      dz^ 


'   dii  ^ 

(2)  (y  -  x)  ( 1  +  .1") '  ^  =  ( 1  +  //-)  - ,  reduces  to 

sin  (<^  —  ^)  rf<^  =  (/^,  if  .>-=taiii9  and  //  =  tau<i. 

(3)  fx^-^  -  y\  =  afl  +  ^^  {X-  +  //-)  %  reduces  to 


dr 


Vr  ( 1  —  ar)      v  a 


-^  =  0,   if  X  =  /•  cos  <^  and  1/  =  r  sin  <^. 


326  INTEGRAL   CALCULUS. 


KEY. 


Page 

Single  equation I.  32G 

System  of  simultaneous  equations VIII.  329 

I.  Involving  ordinar}^  derivatives II.  32G 

Involving  partial  derivatives IX.  329 

II.  Containing  two  variables III.  326 

Containing  three  variables  and  of  first  degree. 

General  form,  P(lx+Q(Jy  +  Rdz  =  0  .     .     .     (36)343 
Containing  more   than    three  variables  and  of 
the  first  degree.     General  form,  Pdxi  +  Qdx2 
+  Rdxs+  —  =0 (37)344 

III.  Of  first  order IV.  326 

Not  of  first  order VII.  328 

IV.  Of  first  degree.    General  form,  J/(/.t  + -V(/// =  0        V.  326 
Not  of  first  degree VI.  327 

V        Of  first  degree.     General  form,  3fdx -{- Xdy 

=  0. 

Variables  separated  or  separable ;   that  is,  of 

or   reducible   to    the    form    Xdx  -f  Tdif  =  0, 

wliere  X  is  a  function  of  x  alone,  and  Y  is  a 

function  of  ?/ alone* (1)  330 

M  and  iV  homogeneous  functions  of  x  and  y  of 

the  same  degree (2)  330 

*  Of  course,  A'  and  Y  may  be  constants. 


KEY.  327 

Of  the  form  {ax  -f  h>/-\-c)  dx  +  {a'x  +  h'y-\-c')'hj 


=  0 


(.3)  330 


Linear.     General  form,  -^  -\- XiV  =  X-,,  where 
(Ix 

Xi  and  X,  are  functions  of  a;  alone  *    ...       (1)  330 

Of  the  form  ^  +  Xjy  =  X^y'\  where  Xj  and  X 
dx 

are  functions  of  .r  alone  * (5>  331 

Mdx  +  Xdy  an  exact  differential.     Test,  D^M 

=  D,N (Gj  331 

Mx  +  Ny  =  0 (7)  331 

Mx-Ny  =  Q (.s)  331 

Of  the  form  F,  (.r?/)  ydx  +  F.  (xy)xdy  =  0     .  (H)  331 

D,M-D,N^  ^  fuuetion  of  x  alone     .     .     .  (10)  331 

N 

^'^~  Dy^^^  a  function  of  y  alone     .     .     .  (11)332 

M 

D^M-  D,N^  ^  function  of  (xy) (12)332 

Ny  —  Jfx 
A  solution  in  the  form  of  a  series  can  always  be 

obtained .     (13)  332 


VI.       Not  of  first  degree. 

Can  be  solved  as  an  algebraic  equation  in  p, 

where  »  stands  for  — (14)  333 

dx 
Involves    only   one   of    the    variables    and  p, 

where  »  stands  for  — (lo)  333 

dx 
Of  the  first  degree  in  x  and  y ;  that  is,  of  the 
form  xfip  +  yf22)=f3P-,  where  p  stands  for 

^ (IG)  333 

dx  ^  J 

Of  the  first  degree  in  x  or // (17)  334 

Homogeneous  relatively  to  x  and  //     .     .     .     .  (18)  334 

•  Of  cour«t',  A',  ami  A',  may  be  ciiiiHtantM. 


328  INTE(;iiAL   CALCULUS. 

Page 

Of  the  form  F{(fi,if/)  =  0,  where  ^  and  ip  are 

functious  of  X,  ?/,  and  —  ,  such  that  4>  =  a 

^  dx 

\pz=h,  will   lead,  on  differentiation,   to  the 
same   differential    equation    of    the    second 

order (ID)  334 

A  singular  solution  will  answer (20)  334 

VII.       Not  of  first  order. 

Linear,    with    constant    coefficients ;     second 

member  zero* (21)  335 

Linear,    with    constant    coeflicients ;     sec<nid 

member  not  zero* (22)  335 

Of  the  form  (a  +  hx) "  ^-  +  A  (a  +  hx) »  '  ^^ 
dx"  dx"~'^ 

+  •••  -1-  Ly  =  X,  where  X  is  a  function  of 

X  alone  t (23)  337 

Linear  ;  of  second  order  ;  coefficients  not  con- 
stant.  General  form,  ^  +  P^+Oy  =  i2; 
dx^  dx 

P,  Q,  and  R  being  functions  of  .r      ...     (24)  337 
Either  of  the  primitive  variables  wanting    .     .      (25)  339 

Of  the  form  — -  =  X,  X  being  a  function  of 
dx'^ 

.r  alone  t (20)  339 

Of  the  form  — 7=  Y,    Y  being  a  function  of 
dx- 

y  alone  f (27)  340 

Of  theA.rm  j|^==/~^( (28)340 

Of  tiie  form ''"-^  =/•'—? (210  340 

dx''      '  dx"  ^ 
Homogeneous  on  the  supposition  that  x  and 

♦  The  firnt  member  is  BuppoBcd  lo  contaiD  only  thooe  terms  involving  the  dependent 
variable  or  itH  derivatives, 
f  Bee  note,  p.  310. 


KEY.  329 

y  are  of  the  dosrce  1,  -•-  of  the  dctrroe  0, 

^^  of  the  degree  -1,  .•• (;H))-341 

Homogeneous  on  the  supposition  that  x  is  of 
the  degree  1,  y  of  the  degree  ?;,  —  of  tiie 

degree  ?i  —  1,  — ^  of  the  degree  u  —  2,  •••        (.'il)  .'III 

Homogeneous  relatively  to  v,  — ,  ^-^,  •••       .     i'.Vl)  341 
■     dx   c7.»r 

Containing  the  first  power  only  of  the  deriva- 
tive of  the  highest  order (33)  341 

Of  the  form  ^  +  X'^  +  y\'^=  0,  Avhere 
f?.i-           dx         \Ji-^\ 
X  is  a  function  of  x  alone  and  Y  a  func- 
tion of  ?/ alone  *  (34)  342 

Singular  integral  will  answer (35)   342 

VIII.    Simultaneous  ecpiations  of  the  first  order  .     .      (3H)    844 
Not  of  the  first  order   .     .     c (3it)   345 

IX.  All  the  partial  derivatives  taken  with  respect 

to  one  of  the  independent  variables  .     .      .      (10)   345 

Of  the  first  order  and  Linear X.  329 

Of  the  first  order  and  not  Linear  .  .  .  .  XI.  329 
Of  the  second  order  and  containing  tlie  deriv- 
atives of  the  second  order  only  in  the  first 
degree.  General  form  RD^'z  -f-  ST),  I>^z  -}- 
TD^^z=V,  where  R,  S,  T,  and  V  may  be 
functions  of  x,  y,  z,  IJ^z,  and  D^z    .     .      .      (!.'>)    .347 

X.  Containing  three  variables (II)   340 

Containing  more  than  three  varial>les  .      .      .      (12)    .34(1 

XL   Containing  tinve  variables (43)    'MG 

Containing  more  than  three  variables  .      .     .      (44)   347 

•  Sec  note,  p.  310. 


330  INTEGIIAL   CALCULUS. 

(1)  Of  or  reducHjle  to  the  form  Xdx  +  Ydy  =  0,  where  X  is 
a  function  of  x  alone  and  I'is  a  function  of  y  alone. 

Integrate  each  term  separately,  and  write  the  sum  of 
their  integrals  equal  to  an  arbitrary  constant. 

(2)  M  and  X  homogeneous  functions  of  x  and  y  of  the 
same  degree. 

Introduce  in  place  of  y  the  new  Aariable  v  defined  by 
the  equation  y  =■  vx,  and  the  equation  thus  obtained  can 
be  solved  by  (1).     * 

Or,  nuiltii)ly  the  equation  through  by ,  and  its 

^  ^  ^  ^      -^  Mx-\- Xy 

first  member  will  become  an  exact  differential,  and  the 

solution  may  be  obtained  by  (6). 

(3)  Of  the  form  {ax  +  by  +  c)  dx  +  (a'x  +  b'y  +  c')  dy  =  0. 
If  ab'—  a'b  =  0,  the  equation  may  be  thrown  into  the 

form  (ax  +  by  +  c)  dx -\ — (ax  +  by +  c)dy  =  0.     If  now 

z  =  ax-{-  by  be  introduced  in  place  of  either  x  or  y,  the 
resulting  equation  can  be  solved  by  (1). 

If  ab'  —  a'b  does  not  equal  zero,  the  equation  can  be 
made  homogeneous  1))'  assuming  x  =  x'—  a,  ?/  =  y'—fi,  and 
determining  a  and  (S  so  tliat  the  constant  terms  in  the  new 
values  of  3/  and  X  shall  disappear,  and  it  can  then  be 
solved  by  (2). 

(4)  Linear.     General  form    --\-  Xiy=  X.2^   where   X,   and 

dx 
Xj  are  functions  of  x  alone. 

Solve  on  the  supposition  that  X2  =  0  by  (1);  and  from 
this  solution  obtain  a  value  for  y,  involving  of  course  an 
arl)itrary  constant  C.  Substitute  tliis  value  of  y  in  the 
given  equation,  regarding  C  as  a  variable,  and  there  will 
result  a  differential  equation,  invohnng  C  and  x,  whose 
solution  by  (1)  will  express  C  as  a  function  of  x.  Sub- 
stitute this  value  for  C  in  the  expression  already  obtained 
for  y,  and  the  result  will  be  the  required  solution. 


KEY.  381 

(5)  Of  the  form  -•  +  XiV  =  X. y",  wIktc   X,  aud   X,  arc 

dx 

functions  of  x  alone. 

Divide  through  by  y",  and  then  intnKhiee  2=?/'""  in 
place  of  ?/,  and  the  equation  will  become  linear  and  may 
be  solved  by  (4). 

(6)  Mdx  +  my  an  exact  differential.     Test  D^  M  =  D,  N. 
Find  I  Mdx,  regarding  y  as  constant,  and  add  an  arl>i- 

trary  function  of  y.  Determine  this  function  of  y  by  the 
fact  that  the  differential  of  the  result  just  mentioned,  taken 
on  the  supposition  that  x  is  constant,  must  equal  Ndy. 
Write  equal  to  an  arbitrary  constant  the  |  Mdx  above 
mentioned  plus  the  function  of  y  just  determined. 

(7)  Mx  +  Ny  =  0. 

Divide  the  first  term  of  Mdx  +  Ndy  =  0  by  3/x,  and 
the  second  by  its  equal  —Ny^  and  integrate  by  (1). 

(8)  Mx  -  Ny  =  0. 

Divide  the  first  term  of  3Tdx  +  Ndy  =  0  by  Mx,  and 
the  second  by  its  equal  Ny,  and  integrate  by  ( 1 ) . 

(9)  Of  the  form  /,  {xy)  ydx  +f^  (xy)  xdy  =  0. 

Multiply  through  bv  ,  and  the   first   mem])t'i 

■'    Mx  —  Ny 

will  become  an  exact  differential.     The  solution  may  then 

be  found  by  (0). 


(10)       ^J^ ^^,  a  function  of  x  alone. 

-^  rngM-n.y  ^^ 

Multiply  the  equation  through  by  e^  s  '  ',  and 
the  first  member  will  become  an  exact  differential.  Pho 
solution  may  then  be  found  by  (G). 


332  INTEGRAL  CALCULUS. 

(11)  — ^ -—-^ — ,  a  function  of  y  alone. 

Multiply  the  equation  through  by  e^  m  '"^^  and 
the  first  member  will  become  an  exact  differential.  The 
solution  may  then  be  found  by  (6). 

(12)  — * i — ,  a  function  of  (xy). 

Multiply  the  equation  through  by  e*^  Ny-Mx  '  "  where 
V  =  xy,  and  the  first  member  will  become  an  exact  differ- 
ential.    The  solution  may  thus  be  found  by  (6). 

(13)  A  solution  oiMdx  +  Ndy=^Q  in  the  form  of  a  series 
can  always  be  o])tained. 

Throw   the   given  equation  into  the  form    -M  =  —  - — 

clx    J     X 
then    differentiate,    and    in    the    result   replace    —     by 

—  — ,    thus    obtaining  a   value  of  ^^    in    terms    of    x 

Jy  dxr 

and  y ;    b}-  successive  differentiations  and  substitutions 

d^  11    d^  ?/ 
get  values  of  — -i,  —4,  etc.,  in  terms  of  a;  and  y. 
dxr    dx* 

If  2/0  is  the  value  of  y  corresiwnding  to  any  chosen 
value  .To  of  X,  y  can  now  be  developed  by  Taylor's 
Theorem. 

We  have  y  =fx  =/(•>•„  -}-  x  —  X(,) 


Xo  +  - 


=  /»•„  +  {X  -  x,)f%  +  (•^-^•o)>r,„  +  i^^ofj^nr 

dxo  2 !        dx„-  3 !        d.V 

where  ^,   ^,   ^,  etc., 

dxQ     dxo^     rfa'o" 

are  obtained  by  replacing  x  and  y  by  Xq  and  y^  m  the 

values  of  ,        ,2        -, 

dy^    (Py^    c^^  gj^^^ 

dx'    dx^'    djr'' 
described  al)ove. 


KEY.  333 

In  the  general  case  ?/u  •»  entirely  arbitrary,  and  if  the 
given  equation  is  at  all  complicated,  the  solution  is  apt  to 
be  too  complicated  to  be  of  much  service.  If,  however, 
in  a  special  problem  the  value  of  y  corresponding  to  some 
value  of  X  is  given,  and  these  values  are  taken  as  //o  and 
iCu,  the  solution  will  generally  be  useful. 

(14)  Can  be  solved  as  an  algebraic  equation  in  j),  wlu-re  p 

stands  for  -^. 
dx 

Solve  as  an  algebraic  equation  in  j),  and,  after  trans- 
posing all  the  terms  to  the  first  member,  express  the  fust 
member  as  the  product  of  factors  of  the  first  order  and 
degree.  Write  each  of  these  factors  separately  equal  to 
zero,  and  find  its  solution  in  the  form  V—  c  =  0  by  (V.) . 
Write  the  product  of  the  first  members  of  these  solutions 
equal  to  zero,  using  the  same  arbitrary  constant  in  each. 

(15)  Involves  only  one  of  the  variables  and  p,  where  2>  stands 

for^. 
dx 
By  algebraic  solution  express  the  variable  as  an  expli- 
cit function  of  j>,  and  then  differentiate  through  relatively 
to  tlie  other  variable,  regarding  })  as  a  new  variable  and 

remembering  that  — ^  =  -.      There  will  result  a  differen- 

dy     p 
tial  equation  of  the  first  order  and  degree  between  the 
second    variable    and    p   which   can    be    solved    by   (1). 
Eliminate  j)  between  this  solution  and  the  given  equation, 
and  the  resulting  equation  will  be  the  required  solution. 

(16)  Of  the  form  xf^p  -f  yf.,p  =f^p,  where  p  stands  for  ' j!.. 

Differentiate  the  equation  relatively  to  one  of  the  vari- 
ables, regarding  p  as  a  new  variable,  and,  with  the  aid  of 
the  given  equation,  eliminate  the  other  original  variable. 
Tiiere  will  result  a  linear  dilf'erential  equation  of  th*-  first 


334  INTEGRAL   CALCULUS. 

order  between  p  aud  the  remaining  variable,  which  may  be 

simplified  by  striking  out  any  factor  not  containing  -^  or 

-^,  and  can  be  solved  bv  (4).     Eliminate  p  between  this 

dy 

solution  and  the  given  equation,  aud  the  result  will  be  the 

required  solution. 

(17)  Of  the  first  degree  in  x  or  y. 

The  equation  can  sometimes  be  solved  by  the  method  of 
(If)),  differentiating  relativel}'  to  the  variable  which  does 
not  enter  to  the  first  degree. 

(18)  Homogeneous  relatively  to  x  and  y. 

Let  y  =  vx,  and  solve  algebraically  relatively  to  p  or  -y, 

p  standing  for  -^.     The  result  will  be  of  the  form  p  —  fv, 

dx 
or  v  =  Fp.     If 

J.     dy      ~     d(vx)       ~        dv  ,  ^ 

dx  dx  dx 

an  equation  that  can  be  solved  by  (1).     If 

'o  =  Fp,  y-  =  Fx>,  y  =  xFp, 

X 

an  equation  that  can  be  solved  by  (16). 

(lU)       Of  the  form  F((f},  if/)  =  0,  where  (f>  aud  ij/  are  functions 

of  X,  y,  and  ^,  such  that  <j>  =  a  and  U/  =  b  will  lead,  on 

dx 
differentiation,  to  the  same  differential  equations  of  the 
second  order. 

Eliminate  —  between  </>  =  a  and  if/  =  b,  where  a  aud  b 
dx 

are    arbitrar}'    constants    subject    to    the    relation    that 
F{a,  b)  =  0,  and  the  result  will  be  the  required  solution. 

(20)       Singular  solution  will  answer. 

Let  -^=2i,  and  express  2^  as  an  explicit  function  of  x 

d 
and   ?/.      Take    — ,    regarding   x   as   constant,    and   see 
^  dy        ""         ° 


kf:y.  385 

whether  it  can  be  made  infiuite  by  writing  eijual  to  zero 
any  expression  involving  y.  If  so,  and  if  the  eciuatiou 
thus  formed  will  satisfy  the  given  differential  equation,  it 
IS  a  singular  solution. 


<,') 


Or  take  —^J-L,  regarding  y  as  constant,  and  see  whether 

it  can  be  made  infinite  by  writing  equal  to  zero  any  ex- 
pression involving  x.  If  so,  and  if  the  equation  thus 
formed  is  consistent  with  the  given  equation,  it  is  a 
singular  solution. 


(21)  Linear,   with   constant   coetficieuts.       Second   member 
zero. 

Assume  y  =  e'"^  \  m  being  constant,  substitute  in  the 
given  equation,  and  then  divide  through  by  e"".  There 
will  result  an  algebraic  equation  in  m.  Solve  this  equa- 
tion, and  the  complete  value  of  y  will  consist  of  a  series 
of  terms  characterized  as  follows :  For  every  distinct 
real  value  of  m  there  will  be  a  term  Ce"" ;  for  each  pair 
of  imaginary  values,  a-f6V  — 1,  a  —  6  V  —  1,  a  term 
Ae"'  cos  hx  +  Be"'  sin  hx  ;  each  of  the  coefficients  A,  B,  and 
C  being  an  arbitrary  constant,  if  the  root  or  pair  of  roots 
occurs  but  once  ;  and  an  algebraic  jjolynomial  in  x  of  the 
(r— l)st  degree  with  arbitrary  constant  coetlicieuts,  if 
the  root  or  i)air  of  roots  occurs  /•  times. 

(22)  Linear,  with  constant  coellleieuts.     Second  member  not 
zero. 

(a)  If  a  particular  solution  of  the  given  equation  can 
be  obtained  by  inspection,  this  value  plus  the  value  of »/ 
detained  by  (21)  on  the  hypothesis  that  the  second  mem- 
ber is  zero,  will  be  the  complete  value  of  tlie  dependent 
variable. 


336  INTEGRAL   CALCULUS. 

(b)  If  the  second  member  of  the  given  equation  can 
be  got  rid  of  by  differentiation,  or  by  differeutiatiou  and 
elimination  between  the  given  and  the  derived  equations, 
solve  the  new  differential  equation  thus  obtained,  by  (21), 
and  determine  the  su[)erfluous  arbitrary  constants  so  that 
the  given  equation  shall  be  satisfied. 

In  determining  these  superfluous  constants,  it  will 
generally  save  labor  to  solve  the  original  equation  on 
the  hyi)othesis  that  its  second  member  is  zero,  and  then 
to  strike  out  from  the  preceding  solution  the  terms  Avhich 
are  duplicates  of  the  ones  in  the  second  solution  before 
proceeding  to  differentiate,  as  from  the  nature  of  the  case 
they  would  drop  out  in  the  course  of  the  work. 

(c)  If  the  given  equation  is  of  the  second  order,  solve 
on  the  hypothesis  that  the  second  member  is  zero, 
by  (21),  obtain  from  this  solution  a  simple  particular 
solution  by  letting  one  of  the  arbitrary  constants  equal 
zero  and  the  other  equal  unity,  and  \ety  =  v  be  this  last 
solution  ;  then  substitute  vz  for  y  in  the  given  equation  ; 
there  will  result  a  differential  equation  of  the  second  order 
between  x  and  z  in  wliich  the  dependent  variable  z  will  be 
wanting,  and  which  can  be  completely  solved  by  (25). 
Substitute  the  value  of  z  thus  obtained  in  y  =  vz  and 
there  will  result  the  required  solution  of  the  given  equa- 
tion. 

(d)  Solve,  on  the  hypothesis  tliat  the  second  member 
is  zero,  and  obtain  the  comiilete  value  of  y  by  (21). 
Denoting  the  order  of  the  given  equation  by  n,  form  the 

n—  1    successive    derivatives    -ii,    — -1 ... ±.       Then 

dx    f?.x~'       dx"~^ 

differentiate  y  and  each  of  the  values  just  obtained,  re- 
garding the  arl)itrary  constants  as  new  variables,  and 
sul)stitute  the  resulting  values  in  the  given  equation  ;  and 
by  its  aid,  and  that  of  the  v  —  1  equations  of  condition 
formed  liv  writinir  each  of  the  derivatives  of  the  second  set, 


KEY.  837 

except  the  nth,  equal  to  the  derivative  of  the  same  order  in 
the  first  set,  determine  the  arlntrary  coelileients  and  sub- 
stitute their  values  in  tlie  original  expression  for  >/. 

(23)  Of  the  form 

(a  +  bx) "'^  +  A  {a  -{-  bx) '-'  'j"^'  +  •  •  •  +  /.//  =  X, 

where  X  is  a  function  of  r  alone. 

Assume  a-{-bx  =  e',  and  change  the  independent  vari- 
able in  the  given  equation  so  as  to  introduce  t  in  place  of 
X.     The  solution  can  then  be  obtained  by  (22). 

(24)  Linear;    of    second   order;    cocllicieiits    not   constants. 

General  form        '  ■" ''  +  I' '[''  +  Q>/  =  li. 
(Li-  (Ix 

(a)  If  a  particular  solution  y  =  v  of  the  equation 

can  be  found  by  inspection  or  other  means,  substitute 
y=cz  in  the  given  equation,  which  will  then  reduce  to 
the  form 

v'^^  +  (2'l^  +  Pi\'^  =  R, 
dar      \   (Ix  J  .(Ix 

and  can  be  solved  by  (2.")).  Substitute  the  value  of  z 
thus  found  in  y  t'z,  and  the  result  will  Ik-  the  general 
solution  of  the  given  ecjuation. 

(b)  The  substitution  of  y  =  rz  in  tiie  given  equation, 
wliere  v   is  given   by  the   auxiliary  ditTi-rential   ecpiation 

2''-  +  I>r  =  0, 
dx 


INTEGRAL   CALCULITS. 

and  can  be  found  by  (1),  and  should  be  used  in  the 
simplest  possible  form,  will  lead  to  a  differential  equation 
in  z  of  the  form 

which  is  often  simpler  tluui  the  original  equation. 

(c)  The  introduction  of  z  in  place  of  the  i:idc|H'n(lont 
variable  x,  z  being  a  solution  of  the  auxiliary  differential 
equation 

da^         dx 

the  simpler  the  better,  will  reduce  the  given  equation  to 
the  form 

which  is  often  simi)ler  than  the  original  e(piation. 

{d)  If  the  first  member  of  the  given  equation  regarded 
as  an  operation  performed  on  y  Can  be  resolved  into  the 
product  of  two  operations,  tlie  equation  can  always  be 
solved.  The  conditions  of  such  a  resolution  are  the 
following;  let  the  given  equation  be 

dx-         dx 

where  w,  ?',  w,  and  R  are  functions  of  x ;  this  can  be 
resolved  into 

>+")(''£+'')-"=-'^' 

where/),  7,  r,  and  .s  are  functions  of  .r,  if 

/(/;-    ,     \  ,         ,     da 

j)r  =  ?<,  'jr  -{-J)  I 1-  .s  )  =  V,  and  (/s  -j- }>  — •  =  ■<<;; 

\(lx        J  dx 

and  the  values  of  p,  7,  i\  and  s  can  usually  be  ()l)tainod 


KEY. 


bv   inspection.       Wo    have    first    to   solve  p \-<iz  =  R 

bv  (4),  and  then  to  solve  /•  ;  -\- sij  =z  bv  (4). 
dx       ^  .    V   / 

(e)   A  particular  solution  of  the  ecjuatiou 

doer  dx 

can  often  be  obtained  by  assuming  that  y  is  of  the  form 
'S.a^x'",  m  being  an  integer,  substituting  this  value  for  y 
in  the  given  equation,  writing  the  sum  of  the  coefHcients 
of  x'"  equal  to  zero,  since  the  equation  must  be  identically 
true,  and  thus  obtaining  a  relation  between  successive 
coefficients  of  the  assumed  series.  The  simplest  set  of 
values  consistent  with  this  relation  should  be  substituted 
in  the  assumed  value  of  y,  which  will  then  be  a  particular 
solution  of  the  equation.  If  this  solution  oan  be  ex- 
pressed in  finite  form,  the  comi)lote  soluti(  n  of  the  given 
equation  can  be  obtained  from  it  by  the  method  described 
in  (24)  (a).  If,  however,  two  different  particular  solu- 
tions can  be  found  by  the  method  just  described,  each 
of  them  should  be  multii)lied  l)y  an  arbitrary  constant,  and 
the  sum  of  these  products  will  be  the  complete  solution 
of  the  given  equation. 

(25)  thither  of  the  primitive  variables  wanting. 

Assume  z  equal  to  the  derivative  of  lowest  order  in  the 
equation,  and  ex[)ress  the  equation  in  terms  of  z  and  its 
derivatives  with  respect  to  the  primitive  variable  actually 
present,  and  the  order  of  the  resulting  equation  will  be 
lower  than  that  of  the  given  one. 

(26)  Of  the  form       •   =  X.      X  boinir  a  function  of  .r  alone. 

dx" 

Solve  by  integrating  n  times  successively  with  regard 
to  0-. 

Or  solve  I)y  (22). 


340  INTEC.HAL   CALCULUS. 

(27)  Of  the  form  —  =  Y.      1' being  a  function  of  y  alone. 

dx" 

^lultiplv  1)V  2'-   and  integrate  relatively  to  a;.     There 
dx  / 7  \2  /• 

will    result    the    equation    r-^j  =-(  Ydy-\-C,   whence 

-^=  (2  j  Ydy -\-  C)2,  an    equation  that    may    be  solved 
dx         J 

by  (1). 

(28)  Of  the  form  ^/=/f^. 

dx"         dx"-^ 

Assume 

d"  '//  ^,        dz       .  ,        dz  rdz  ,  ^ 

^^  =  z,  then  —  =  fz  or  dx  =  — ,  a;  =  I \-C. 

dx"  '  dx     '  fz  J  p 

After  effecting  this  integration,  express  z  in  terms  of  x 

and    (7.      Then,    since   z  =  '^^,    'I^-'  =  F(x,   C),    an 
dx"  '      dx"^^ 

equaticjii  that  may  be  treated  by  (2()). 
Or,  since 

d^  =  ,/ll!y=  (;./.,  +  ,  =  f!^  +  0,  since  dx  =  '^. 
Jx"  '  dx"-'     J  J  fz  fz 

Continue  this  process   until  y   is   cxpn-ssed    in    ti'rnis   of 
z  and  //  —  1,  arl)itrary  constants,  :uid  Ihcu  t'liminati'  z  l)y 

the  aid  of  tiie  eiiuation  x=  (  ^ -\- 
J  fz 


C. 


Let  — — •   =  2,  and  the  ecination  becomes        ,=/^i  and 
dx"  •  fix- 

may  be  solved  l»y  (27). 


KEY.  841 

(30)        Homogeneous  ou  the  supposition  that  x  and  y  are  of  the 

degree  1,  _^'  of  the  de<>i-ee  0,      --  of  the  degree  —  1,  •••. 
ax  "  (/.r 

Assume  x  =  e^,  y  =  e^z,  and  by  clianging  the  varialUes 

introduce  6  and  z  i^to  the  equation  in  the  place  of  x  and  >/. 

Divide  through  by  e*  and  there  will  result  an   equation 

involving  onlv  z,     -,  — -,  •••,    whose  order   iii:iv   lie  de- 

(16    de- 
pressed by  (2.3). 


(31)       Homogeneous  on  the  supposition  that  x  is  of  the  degree 

1,  7/  of  the  degree  ?(,     ''  of  the  degree  ?i  —  1,  -  •''  of  the 

clx  rf.ir 

degree  /i  —  2,  •••. 

Assume  x  =  e^,  y=ze"^z,  and  by  changing  the  variables 
introduce  0  and  z  into  tlie  equation  in  the  place  of  x  and  //. 
The  resulting  equation  may  be  freed  from  6  by  division 
and  treated  by  (25). 


(32)        Homogeneous  relativelv  to  ?/,  ~,   — -,•••• 

dx    dxr 

Assume  i/  =  e',  and  substitute  in  the  given  equation. 

Divide  through  by  e*  and  treat  by  (25). 


(33)        Containing  the  first  power  only  of  tiie  derivative  of  the 
highest  order. 

The  equation  may  be  exnrt. 

Call  its  first  member — .    H  n  is  tlie<jrdfr  ofthe  filiation, 

represent  '^ ~  by  j>  and  _  -^  by  ^  .     Multiplv  th*-  it-rni 

'  (/.»;"-'      ^  '  dx"     '   dx 

containing  '  ^'  h\  <lx  and  integrate  it  as  if  j>  were  the  onlv 

''■''     '  d"   'v 

variable,  calling  the  result  T', ;   tiien  replacing /i  by    .^~, 


342  INTEGRAL   CALCULUS. 

(Ijr 
fiud  the  complete  derivative        ',  and  form  the  expression 

— -',  representing  it  hv ^.     U  - — ?  contains  the 

dx       dx  "    dx  dx 

first  power  only  of  the  highest  derivative  of  ?/,  it  may 

itself  be  an  exact  derivative,  and  is  to  be  treated  pre- 

dV 
ciselv  as  the  first  member  of  the  given  equation  —  has 

dx 

been.     Continue  tliis  process  until  a  remaintiier  '^  of 

dx 
the  first  order  occurs. 

Write  this  equal  to  zero,  and  see  if  the  equation  thus 
formed  is  exad^  see  (G).  If  so,  solve  it  by  (6), 
throwing  its  solution  into  the  form  F„_i  =  C  A 
complete  first  integral  of  the  given  equation  will  be 
Vi-\- Ui+ ■■■  +  V„i=  C.     The  occurrence  at   any  step 

of  the  process  of  a  remainder  — ^,  containing  a  higher 

power  than  the  first  of  its  highest  derivative  of  y,  or  the 
failure  of  the  resulting  equation  of  the  first  order  above 
described  to  be  exacts  shows  that  the  first  member  of  the 
given  equation  was  not  an  exact  derivative,  and  that  this 
method  will  not  apply. 

(34)       Of  the  form  '^^x'^+ y\'^;''^  =0,  where   X  is  a 
(/ar  dx         L"-'J 

function  of  x  alone  and  Y  a  function  of  y  alone.     jNIultiplv 

'[: 

and  may  be  solved  by  (33). 


through  by  [  -  •'  |      and  the  eciuation  will  become  exact, 


(3;"))       Singular  integral  will  answer. 

Call ^9),  and      ^  7,  and  find  ^,  regarding  «  and  o 


da;""'  dx"  dp 


dq 


as  the  onl}'  variables,  and  see  whether  ~J-  can  be  made 

d}) 
infinite  by  writing  i-qtial  to  zero  any  factor  containing  p. 


343 


If  so,  oliiniuato  7  hctwoon  this  equation  and  the  pjivon 
♦.'(luation,  and  if  the  result  is  u  sohitiun  it  will  be  a  singular 
iute<rral. 


( 3(; )        General  form,  Pdx  +  Qfly  -\-  R<h  =  0. 

U  the  e{iuation  can  hi'  redueed  to  the  form  Xdx  +  Yd;/ 
+  Zdz  =  0,  where  X  is  a  function  of  .r  alone,  Y  a  function 
of  //  alone,  and-  Z  a  function  of  z  alone,  inte{2:rate  each 
term  separately,  and  write  the  sum  of  the  integrals  equal 
to  an  arbitrary  constant. 

If  not,  integrate  the  equation  by  (V.)  on  the  supposition 
that  one  of  the  variables  is  constant  and  its  ditlerential 
zero,  writing  an  arbitrary  function  of  that  variable  in  place 
of  the  arbitrary  constant  in  the  result.  Transpose  all  the 
terms  to  the  first  member,  and  then  take  its  complete 
differential,  regarding  all  the  original  variables  as  variable, 
and  write  it  equal  to  the  first  member  of  the  given  equa- 
tion, and  from  this  equation  of  condition  determine  the 
arbitrary  function.  Substitute  for  the  arbitrary  function 
in  the  first  integral  its  value  thus  determined,  and  the 
result  will  be  the  solution  required. 

If  the  equation  of  condition  contains  any  other  varia- 
bles than  the  one  involved  in  the  arliitrary  function,  they 
must  ])e  eliminated  by  the  aid  of  the  primitive  equation 
already  oV)tained ;  and  if  this  elimination  cannot  be  per- 
formed, the  given  equation  is  not  derival)le  from  a  single 
primitive  equation,  l)ut  must  have  come  from  two  simul- 
taneous primitive  equations. 

In  that  case,  assume  any  arbitrary  ('(luation /(.r,//-^)  =0 
as  one  primitive,  differentiate  it,  and  eliminate  betwei-n  it 
its  derived  equation  and  the  given  equation,  one  variable, 
and  its  differential.  There  will  result  a  ditfeiential  eijua- 
tion  containing  only  two  varialHcs,  which  may  be  solvrd 
by  (III.),  and  will  had  to  the  scccjiid  [jrimilive  of  tlie 
given  equation. 


344 


INTEGRAL   CALCULUS. 


(37)       Gonoral  form,  Pdxi  +  Qdx.,  +  Rdx^  + =  0. 

If  the  ociuation  can  be  reduced  to  the  ronn  XidXi-{- XMxj 

+  X^dx^  + =  0,  where  Xi  is  a  function  of  Xi  alone.  X., 

a  function  of  x,  alone,  X,  a  function  of  a-g  alone,  etc.,  inte- 
grate each  term  separatel}',  and  write  the  sum  of  their 
integrals  equal  to  an  arbitrary  constant. 

If  not,  integrate  the  equation  by  (V.),  on  the  supposi- 
tion that  all  the  variables  but  two  are  constant  and  their 
differentials  zero,  writing  an  arbitrary  function  of  these 
variables  in  place  of  the  arbitrary  constant  in  the  result. 
Transpose  all  the  terms  to  the  first  member,  and  then 
take  its  complete  differential,  regarding  all  of  the  original 
variables  as  variable,  and  write  it  equal  to  the  first  mem- 
ber of  the  given  equation,  and  from  this  equation  of  con- 
dition determine  the  arbitrary  function.  Substitute  for 
the  arbitrary  function  in  the  first  integral  its  value  thus 
determined,  and  the  result  will  be  the  solution  required. 

If  the  equation  of  condition  cannot,  even  with  the  aid 
of  the  primitive  equation  first  obtained,  be  thrown  into  a 
form  where  the  complete  differential  of  the  arbitrary  func- 
tion IS  given  equal  to  an  exact  differential,  the  function 
cannot  be  determined,  and  the  given  equation  is  not  deriv- 
able from  a  single  primitive  equation. 

(3H)        System  of  simultaneous  equations  of  the  first  order. 

If  any  of  the  equations  of  the  set  can  lie  integrated 
sei^arately  by  (II,)  so  as  to  lead  to  single  primitives,  the 
problem  can  be  simi^lified  ;  for  by  the  aid  of  these  primi- 
tives a  number  of  vanal)les  equal  to  the  number  of  solved 
equations  can  be  eliminated  from  the  remaining  equations 
of  the  series,  and  there  will  be  formed  a  simpler  set  of 
simultaneous  equations  wliose  primitives,  together  with  the 
l)nmitives  already  found,  will  form  the  pnmitive  system 
of  the  given  equations. 

There  must  be  n  ecjuatiDns  connecting  )i  +  1  variables, 
in  order  that  the  s^-stem  ma}-  be  dctcnnniate. 

Let  X,  u;,,  a-j ,  x„  be  the  origmal  variables.     Choose 


KEY.  345 

any  two,  a  and  .r,,  as  the  iiulopcndcnt  and  the  prineipal  de- 
pendent variable,  and  l)y  sneeessive  eliminations  Ibnn  the 

n  e(|nations  -  —  =  f\{j\Xi,.r.,, ,.i-„) ,  ^  =  f.{j\.i\,.v.,, , j-„), 

(Lv      '  '  dx 

up  to  '^'=/.(.r,.ri..r.,, .r„).     Difierentiate  the  lirst 

of  these  with  respect  to  x  n  —  1   times,  substituting  for 

— ?,  -^, ,-^'.  after  each  step  their  values  in  terms  of 

dx     dx  dx 

the  original  variables.  There  will  result  »  equations, 
which  will  express  each  of  the   n   successive  derivatives 

dx,     d'x,    d^x,  d"x,     .      .  .. 

— -',    — -',  -',  , ',   HI   terms  of  .r.   .r,,  x., x„. 

dx      dxr     djr  dx" 

Eliminate  from  these  all  the  variables  except  .r  and  x,, 
obtaining  a  single  equation  of  the  »th  order  between  x 
and  X,.  Solve  this  b}'  (VII.),  and  so  get  a  value  of  .r,  in 
terms  of  x  and  n  arbitrary  constants.     Find  by  ditleren- 

dx    d-  r  d"~^x 

tiating  this  result  values  for  -^,  — ^', , ^/,  and  write 

dx     dx-  dx"  ' 

them  equal  to  the  ones  already  obtained  for  them  in  tenns 
of  the  original  variables.  The  »  —  1  equations  thus  formed, 
together  with  the  equation  expressing  .f,  in  terms  of  x  and 
arbitrary  constants,  are  the  complete  primitive  system 
required. 

(3'J)  System  of  simultaneous  equations  not  of  the  first  order. 
Regard  each  derivative  of  each  dependent  variable, 
from  the  first  to  the  next  to  the  highest  as  a  new  vaiiable, 
and  the  given  equations,  together  with  the  equations  de- 
fining these  new  variables,  will  fonn  a  system  of  sinuilta- 
neous  equations  of  the  first  order  which  may  lx>  solved  by 
(;>.S),  Eliminate  the  new  variables  representing  the 
various  derivatives  from  the  ecjuations  of  the  solution,  and 
tlie  equations  obtained  will  be  the  complete  primitive  sys- 
t.'m  recjuired. 

(40)        All  the  partial  derivntives  taken  with  respect  to  one  of 
the  independent  variables. 


346  INTEGRAL   CALCULUS. 

Integrate  by  (II.)  as  if  that  one  were  the  only  indepen- 
dent variable,  replacing  each  arbitrary  constant  by  an 
arbitrary  function  of  the  other  independent  variables. 

^^41)       Of  the  first  order  and  linear,  containing  three  variables. 

General  form,  l*I)^z  +  QD^z  =  R. 

Form  the  auxiliary  system  of  ordinary  differential  equa- 
W/j*       (111       (Tz 
tions  —  =  -^  =  — ,  and  mtegrate  by  (38),     Express  their 

primitives  in  the  form  u  =  a,  v  =  b,  a  and  b  being  arbi- 
trary constants  ;  and  ?<  =/y,  where /is  an  arbitrary  func- 
tion, will  be  the  required  solution. 

(42)  Of  the  first  order  and  linear,  containing  more  than  three 

variables.      General    form,    P^Dx^z  -f  P^Dx^z  -f- =  i?, 

where  a;,,  a-g,  ,  cc„,  are  the  independent  and  z  the  depen- 
dent variables. 

Form  tlie  auxiliary  system  of  ordinary  differential  equa- 
tions ^L^J:^^}^ =  ^'  =  1?5,  and  integrate  them  by  (38) . 

Ex])ress  their  primitives  in  the  form  v^  =  a,  u.  =  &,  Vg  =  c, 
and  Vj  =■  f{^V).2^v^-, ,v„),  where /is  an  arbitrary  func- 
tion, will  be  the  required  solution. 

(43)  Of  the  first  order  and  not  linear,  containing  three  varia- 
bles, F(a;,?/,2;,/),7)  =0,  where  p  =  D^z^  q  =  D^z. 

Express  7  in  terms  of  cc,  y,  z  and  p  from  the  given  equa- 
tion, and  substitute  its  value  thus  obtained  in  the  auxil- 

lary  system  of  ordinary  differential  equations  _^  =  ^V 
= ; —  =  — .     Deduce  by  integration  from 

the.se  equations,  by  (30),  a  value  of  p  involving  an  arbi- 
trary constant,  and  substitute  it  with  the  corresponding 
value  of  7  in  the  equation  dz  =  pdx  -f  qdy.  Integrate 
this  result  l>y  (3G),  if  possible;  and  if  a  single  primitive 
equation  be  obtained,  it  will  be  a  complete  primitive  of  the 
given  equation. 


KEY.  :)[! 

A  sinn:iilar  solution  ma}'  be  ohtainod  by  finding  tiie 
partial  derivatives  D^z  and  D^z  from  the  given  equation, 
writing  them  separately  equal  to  zero,  and  eliminaliiig  j) 
and  q  between  them  and  the  given  equation. 

(44)  Of  the  first  order  and  not  linear,  containing  more  tlian 

three  variables.    F{Xi,x., ,  .t„,  z.p^.p.,, ,p,^)  =  (J,  wla're 

Pi  =  Dx,z,  2h=J^x,z. 

Form  the  linear  partial  ditTcrential  equation  5j[(Z).r(F 
+  PiD.F)Dp<i>  -  Dp.FiDj:.<P+2HDz^)]  =  0,  where  *  is 

an  unknown  function  of  (x'l,  >^',„i?ii iPn)i  ^'^f^  where 

2,-  means  the  sum  of  all  the  terms  of  the  given  form  that 
can  be  obtained  by  giving  i  successively  the  values  1,  2, 
3, ,  n. 

Form,  b3'(42),  its  auxiliary  system  of  ordinary  differen- 
tial equations,  and  from  them  get,  by  (38),  n  —  1   mte- 

gi-als,  <I>i  =  (fi,  <I>o  =  a.,, ,4>„_i  =  a„_i.    B}'  these  equations 

and  the  given  equation  express  j^i,  j^a^  ■> /'«  in  terms  of 

the  original  variables,  and  substitute  their  values  in  tlie 

equation  ch  =  pi  (/.r,  +  p., dr.,  -f- +p„  fU',..     Integrate  tins 

b}'  (37),  and  the  result  will  be  the  required  complete  primi- 
tive. 

(45)  Of  the  second  order  and  containing  the  derivatives  of 
the  second  order  only  in  the  first  degree.  General  form, 
RD/z  +  SDJ)^z  +  TD^z  =  T",  where  li,  ^',  T,  and  I'may 
be  functions  of  .x',  ?/,  2,  Z>,2,  and  D^z. 

Call  I),z  p  and  D^z  q. 
Form  fu'st  the  ecjuation 

/.',/,/-■  _  Si] .all,  +  T(h-  =  0,  [11 

and  resolve  it,  sup[)osing  the  first  menilier  not  a  complete; 
square,  into  two  efiuations  of  the  fijrm 

(hj  —  )ii ,  il.r  =  0,  d>/  —  m.,(lc  =  0.  [2] 

From  the  first  of  these,  and  from  th<'  equation 

Rdpdii  -I-  Tdqdj-  -  \  'dud'f  =  0,  [3] 


348  INTEGRAL   CALCULUS. 

combined  if  noodful  with  the  equation 

dz  =  pdx  +  qdy, 

seek  to  obtain  two  integrals  ?f,  =  a,  Vi  =  fS.  Proceed- 
ing in  the  same  wa}'  with  tlie  second  equation  of  [2], 
seek  two  otlier  integrals  u.,  =  a„  v.,  =  ^i ;  then  the  first  in- 
tegrals of  the  proposed  ecjuation  will  be 

«<i  =/!«!,        ^h=fiV2,  [4] 

where/,  and/o  denote  arbitrary  functions. 

To  deduce  the  final  integral,  we  must  either  integrate 
one  of  these,  or,  determining  from  the  two  p  and  q  m  terms 
of  «,  ?/,  and  z,  substitute  those  values  in  the  equation, 

dz  =  pdx  +  qdy^ 

which  will  tlicn  become  intcgrable.  Its  solution  will  give 
the  final  integral  sought. 

If  the  values  of  m^  and  mo  are  equal,  only  one  first  in- 
tegral will  l)e  obtained,  and  the  final  solution  must  be 
souglit  b}'  its  integration. 

When  it  is  not  possible  so  to  combine  the  auxiliary 
ecjuations  as  to  obtain  two  auxiliary  integrals  u  =  a,  v  =3, 
no  first  integral  of  the  proposed  equation  exists,  and  this 
metliod  of  solution  fails. 


Ciy 


KEY. 


340 


Ans.  cti 


EXAMPI.KS. 

^^)   sinx  cosy .  dx  —  coso-  siu  v .  dy  =  0.       Ans.  {m%ii  =  c cosx. 

i-(2)    {X  +  yy-'l!l  =  a\  Ans.  y  -  a  tair' ''^^  =  c. 

dx  '  (I 

^)    ^  =  siu(«^-^). 

KS)    (2/_a;)(l+a^)'||-^  =  (l+2/=^)l 

til    ^ (tan  ^^  —  lt\n""'.r)    =  tan  '?/  + 


{1-^1= 


^l«s.  sin"'-  =  0  —  X. 

X 


J' 


vi^ 


Ans.  cti 


^?(s.   2  <(  {.V-  +  y-)  =  {x-  -f-  r)  -  —  ^ 


COSC  + ?/sin( 


(7)    [2  ^(^7/)  —  x]  (?// +  y(/.f  =  0,  ^7is.  y 


■  ce  si-.. 


jj^x-f)  +  '2xn/^^  =  i). 


^ns.  xe*  =  c. 


.V  =  c. 


v";    (9)    (2a;-2/+l)dr  +  (2»/-x-l)f/.y  =  0. 
'     /  Ans.   .r-  —  .ry  +  y-  +  x  —  y 

l/lO)   ^-^  +  y  COS X-  =  i^illi'.  ^.l,,.s.   V  =  sill  ./•  -  1  +  ce-'"'. 

•^  dx  2 

(  1  -  x^)  -^  -  .r.v  =  axy\         Ans.  y  =[cy/{\-  .»- )  -  a]  '. 


1 


12)   XI 

"•^'  Ans.    (.1-  +  //•-)■-  -  2  (r  (r"  -  »/')  =  < 


(12)   a-z/(l+^.'r)^.=  l 


^jis.      =  2  —  y-'  +  ce 


350/  ->       V 


INTEGRAL   CALCULUS. 


-.  {U)l xdx  +  ydy)+  ^Ay^lV^  =  0.        Ans.  ^+^'  +  tan^^^  =  c, 
'  \  '  ■  aj-  +  i/-  2  a; 

yl»s.  {'2y  —  x^  —  c)  [log  (x-  +  y  —  \)  +  x  —  c\  =0. 
'^"^'  (^  ~  I  ~  7  V  ^~~^  (^^»^  - 1  -  c  W  0- 

(19)  ri-?/^-^;y^^Y-2^^+<=o. 

ory  yaxy         x*    ox*      xr 


^y/.5,    2/  +  log  — -^-^ —  ^"     !/  —  log -^  —  c  1  =  0. 

\  y  J\  y 


(20)  y  =  ic^^  -(- 1:^  _  T-l^ )  .  Ans.  y  =  ex  +  c  — 


y^)  y 


dX  dX  yj^^^y  .     •   _U    1   V 

f\   ^  p       /.  i  Singular  solutiou,  y  =^ '—■ 

4 

1)  2/  =  2/   t~     +2^*/-      /^     ,  ^ns.  ?/-  =  2ca;  +  c-. 

rZxv  dx         M./l* 


'^^>  [>-(:!!)>= <^-'-«=);;: 


L^^O 


<->-^-t+<;;! 


jifff'f 


(24)  ar^v'    +^-.V   •'+^■''  =  0 


'// 


Urr-^-\/x- 


Ans.  7/2  -  c.r-  --I-  -^^-^  =  0. 
1+c 

^1/iS.  y-  =  2ca;  +  c^. 
^Hs.  p-  +  cry  +  a^.T  =  0. 


,  ^f-Ci 


351 


(25)  / 

(26)  X 


Ans.    {b  +  i/Y  =  4  ax,  /(a)  -f  6  =  0. 


,/- -  x>/'^''~\-       Ans.  -^  =  6,  r(a)  +  b  =  0. 
"  dx  y  —  a 


dx 


xlns.  y  =  ((\,  +  c,  X  +  Ca^r  +  Cj  ar')  e'. 

^bis;  y^-  +  {A  +  Bx)  cosx  +  (C+  />a;)  sino,-. 

(30)   ^_2^  +  4v  =  f?'cosx. 
dar'         d.i- 

yl/(,s.  y  =  Ae  ^  +  el  (J5—  "^   Jcosa;+^C  +  ]-^Jsina-  . 
^^rT"  d*y      ^ d^y      d?y  _    , 


d-t;-"         dor^      du? 


Ans.  y=  {A  +  Bx)  e'  +  {C  +  Dx)-\-\)>x-  +  2,.r^  +  |-  + 


•20 


v^<^ 


4f^  +  4^  =  .-^. 
"^*  Ans.  y  =  {A  +  Bx)e-^+\{-2.i-  +  Ax-{-Z). 


(34)   ^  +  4.v  =  a;sin-x 


^Ins.  ?/  =  ^1  cos  a-  +  2J  siu  .c  -f-  -  sin  x. 


Ans.  y=f  A- -^^cos2x  +  (b-  p  \  sin  ^  J:  +  ^ 

^-^/  +  2y  =  xlogx. 
d.r 
Ans.  y  =  ^Ix  cos  (lof?  .r)  +  7>V  sin  (lo<T.r',  -|-  .r  logrx 


(35)   x^S-a:^  +  2y  =  xlogx, 
dx^         dx 


352  INTEGRAL   CALCULUS. 


(36)  ar-'^-a^^-f  2a;'i^-2y  =  ar'4-3a;. 
dx^  (Ix-  dx 


Ans.  y  =  x{A  +  B log x)  +  Car'  +  -  -  3 x fl  +  05>g^^ 

. .  7 )   ''" ''  +  -  '-^  -  »-'  V  =  0.  Ans.  v  =  -  (Ae'"  +  Be  ") . 

.7.'-       X  dx  '  '       X 

ZA — ''        '  ■  Ans.  y  =  A  cos  (sin  x)  +  B  sin  (sinx). 

(39)  {\-x^y'ly,  +  y=.Q. 

dx-  ,  .        . 

Ans.  y  =  VT^^^  (A  +  B  1( )g  ^-^' 

(40)  {\+x')'^-2x'^  +  2y  =  i). 

''•*-■"  ^^"  .'L^s.  y  =  Bx  +  .1(1-  x') . 

,.,-    d-y         X      dy  ,       1  , 

(41 )  ,  —  ^ ^  H y  =  x  —  \. 

dx^       X  —  I    dx      x  —  I  .  i   ,   ,    r>         /  1    .     ^\ 

xhis.  y  =  Ae'  +  Bx  —  {\  +  x^) 

(42)  ^pl,  -  -^a-  (1  +  .^)  'hL^-2{\+  .r)  y  =  x\ 

dx-  dx  „ 

Ans    y  =  A.ve''  +  Bx • 

2 

(43)  siu-.c^^  -2y  =  0.        Ans.  y  =A  ctn.c  -\- B  (l  -  a;ctna;). 


(I-O  ^(+-vi^  ,/  =  .'f-  +  loga-\ 
dx-      X-  log  x  \a;  J 


Aw^.  y  =  A  log  a:  +  e^  log.f  +  J3(  loga;  |  — x 

V        -^  logo; 


/4rN    <i'y      n(         ((\d>f   ,   (  „      ,^n(\\ 

^''^d^-x-^dx^{:'--'-^)^'= 


Ans.    y  =  e"^lA  +  -^,+  "^ 


JT"  '      2  (2a  -  1) 

(46)!''?--^''i'+f,.'+4,y,=o. 

dx-      x  dx      \         x-j        .  ,   .  ,   7)   •        \ 

^  ^       x\ IIS.  y  =  a-  ( A  cos (/x  4  B  sin  ax) . 


KEY.  353 

(47)  g-2to^+6Vy=0. 

Arts.  y  =  e'^{Acosx-\/b  +  Bs\ux\/b). 

(48)f:f-4./'"  +  C4:r--:i)..  =  ,.". 

Alts,  y  =  e'  {Ac^ -{- B*"     —1). 

(49)  (l_ar')g-4.T|J-'^-(l4-ar^)y=.r. 

]  * 

Aiis.   >/  =  {x  +  .1  COS.C  4-  B  sin.'-). 

1  —  X- 

(50)  'i!^--L  ^^•'/4-^±j/^ii^v  =  o. 

rfx^'      ^y^  dx  Axr        ' 

An.^.  y  =  e'''4Ax-  +  ?-\ 

(51 )  ^-^  +  2  »  ctn  //.r-'^  +  (m*  -  ^i^)  y  =  o. 

(/x-  dx     t  ,  .  ,    T>   •         x 

^ln.s.  y  =  (^1  cos  ?<i.r  +  B  siii  »i.i-)  esc  dx. 

(52)  (ar^_l)^  +  a:^_c=^y  =  0. 


Ans.  y  =  A{x  +  \/x'-\y+B(x-y/x^-\y 

(53)    — ^  H -\ — y  =  0.  Ans.   )/  =  A>im    4-ijeos 

dor      X  dx     X*  '  x  x 


(54) 


d'y      3a;  +  l  (/y         f        6(x+])        T_  ^ 
(?x-'       .1-  -  1    f/.r      •   L  ( .r  -  1 )  ( 3  a-  +  5 )  J 


Ans.  2/  =  [.l4-i>'log((x-  iy(3x  +  5))]V(x-  l)f»(3a;  +  5). 

(65)    (l-^,-^'-.-^-c-,v=0. 

Ans.  y=  Ae"'"    '  +  Be''""  ''. 

(5G)    (H-f,a-)^  +  ax^-n2»/  =  0. 
dxr  dx 

Ans.  y  =  A{^l+ax'-^-xy/ay^-hBiy/l+ax'  +  x \fa)  ^. 

(57)    (X  -  1 )  (X  -  2 )  'f  •'  -  ( 2  .r  -  ;^ )  '1^  +  2  .V  =  0. 
d.i"  dx 

Ans.  2/  =  c  (X  -  2)'^  -h  c'  (X  -  2)  [(X  -  2)  log  (x  -  2)  -  1  ]. 


354  INTKcaiAL   CALCULUS. 


(58)  (3  -  X) ''"'?  -  (9  -  4a;)  ^  +  (6  -  3a:)  //  =  0. 

Alls,  y  =  re^  +  c'e^i x -] ar ar). 

(59)  (a''-x-)'^'y-Hx'^-\2j/  =  0. 

'^•^'            '^"^        .                      c        ^   ,(a-  +  3x2) 
Ans.  y  =  - „  +  c- '• 

Ans.  y  =  —  [A  (sin  nx  —  nx  cos  nx)  +  B  (cos  nx  +  yia;  sin  nx)  ] . 

ar 

(61)  ^+li^^  =  0.  ^Hs.  y  =  cloga;  +  c'. 
dx-      a;  dx 

(62)  Ar'i?  +  .^'j/^f?!  +  4 .,/SY+  2:.yf?  =  0. 
\       dx  J  dx-  Y<-y  dx 

Ans.  cr -\- cxy  =  c' X. 

(03)   (x^  +  2,"^yf|  +  2,/*)'+  34^  +  i,  =  0. 

V  dxjdx^  \dxj  dx    t^.    ,      «    *  ■   .         i 

^  ^  ^     -^  iMiid  a  first  integral. 

Ans.  x--=^  +  2/^1  -^  )  +  av/  =  c, 
dx         \dfa;/ 

(64).r-!^+:.f|  +  (2.r,-l)';''  +  ../-  =  0. 

rind  a  first  integral. 

Ans.  X? — -^  —  x-^  +  xy^  =  c. 
d.v         dx 

1    ( G5 )   -^  +    ^y-   +  — ~    =  0.     Ans.  {x-a){>/-b)(z-c)=  c. 
1/      •     X  —JM      y  —  b      z  —c  _ 


6)  -{y +  z)dx -\-dy +  dz  =  0.  Ans.  e'  (y  +  z)  =c. 


(67)^  +  4x4-^  =  0,     ^  +  3y-x=0. 

dt  4  dt  .^  ^ 

,       J  Ana.  x  =  ce  -i -•'      y  =  (ct+Ci)e  2. 

^^-^^^^     .     V 


KEY. 


866 


(68)  ^  +  m'x=0,     ^-vrx  =  0. 


(G9)   D,z=-y—. 
x-\-z 


h(S.  X  =  A  sin  m(  +  B  cos mt,     x  +  y  =  Of  +  D. 
Ans.  e~*  (.c  +  V  +  2)  =  (fii/. 

( 70)  xzD^z  +  i/zD  z  =  Qcy.  Ans.  z-  =  xy  -f-  <f>  (-] 

\yJ 

(71 )  D,z.D^z=l.  Ans.  z  =  a.r  +  -^  +  b. 

a 

(72)  j-D/z-\-->x>/D,D^z  +  yWJ^z  =  0.  Ans.  z  =  x<lify\+tpf^\ 

(73)  (D^zYD/z  -  2  I),zD,zD,D^z  +  {D,z)-  D;z  =  0. 

^l«s.  y  =  x<f>z  +  ij/z. 


(74)   D,z.D,D^z-D^z.DiZ  =  0. 


Ans.  X  =  <l>y  +  {l/z. 


APPENDIX. 


Chap.  V.J  INTEGRATION.  66 


CHAPTER  V. 

INTEGRATION. 

74.  We  are  now  al>le  to  exteiKl  iiuiU'rially  our  list  of  formulas 
for  direct  integration  (Art.  55),  one  of  which  may  l>e  obtained 
from  each  of  the  derivative  formulas  in  our  last  chapter.  The 
following  set  contains  the  most  important  of  these  :  — 

Z>,  log  X  =  -  gives  /,  -  =  log  X. 

'     '='  X  ^  ^^x  "" 

Z),a'  =  a'logc  "  /^a'log((  =  «'. 

D^e'  =  e'  "  f^e^  =  e\ 

D^sin.f  =  cosx  "  /^cos.t  =  sin-c. 

i>,cosa.'  =  —  sinx  "  fA~  sin.r)  =  cosa*. 

Z),logsina;  =  ctnx  "  y'^ctn.f  =  log.sin.r. 

Z>,logcosa;  =  —  tanx  "  f{~  tan.f)  =  logcosx. 

DMn-'x  = ^ — -         -    /;       '  ■=— '• 


/;jan-'.i-=— ! —  "      /;_l_=t:in-'j-. 

1  +  ,'»•='  1  -f-  .'- 

l)^  \ers- '  .r  = '  •      f, —  =  vers" '  j-. 

V(2x--x-')  •    VC-^-^--^) 

The  second,  fifth,  and  seventh  in  the  second  group  can  Ixj 
written  in  the  more  convenient  forms, 


66  DIFFERENTIAL   CALCULUS.  fABT.  76. 


/.«' 


a- 


loga  ' 

f^s\nx=  —  cosx;  . 

y^tanx  =  —  log  cos  X. 

75.  When  tlic  expression  to  be  integrated  does  not  come  under 
any  of  the  forms  in  the  preceding  list,  it  can  often  be  prepared 
for  integration  by  a  suitable  change  of  variable,  the  new  variable, 
of  course,  being  a  function  of  the  old.  This  method  is  called 
integration  by  substitution,  and  is  based  upon  a  formula  easily 

deduced  from  D^ (Fy)  =D^Fy.D^y\ 

which  gives  immediately 

Fy=f^{D^Fy,D^y). 


Let 

u=D,Fy, 

then 

Fy=f,u, 

and  we  have 

f,n=f{uD^y)', 

or,  interchanging  x  and  y, 

Ln=f,{uD,x).  [1] 

For  example,  required       f{a  +  bxY. 


by  [1] ; 


Let 

z  =  a  +  bx, 

and  then 

f{a  +  bxy  ^/^z"  =f{z"  .  D,x) 

but 

H-l 

--h 

hence  Ma  +  bxy  =  lrz'^=].  ?^. 

6  6  n-l-  1 


Chap.  V.]  INTEGRATION.  67 

Su])stituting  for  z  its  value,  we  have 

,;(„+,,.,.).=i(i±M:::. 

P^XAMPLK. 

Fiiiil  /;  — ?—  .  Ana.    i log(a  +  /-/J-)  • 

a  ■\-hx  h 

7(!.    If /;<•  represents  a  funetioii  that  ean  be  integrated, /((i  +  ^j*) 
can  always  be  integrated  ;  for,  if 

2  =  a  -f-  ^^1 

then  D,x  =  - 

b 

and  fj{a  +  hx)  =  fjz=  f,  fzD,  x=-l\fz. 

b 


EXAMPLES. 

Find 

(1)   f^smax. 

^-Ih.s. cos  ax. 

a 

(2)   y;  cos  ax. 

4                  1       • 

Ans.    -sin«jr, 
a 

(3)   /.tanax. 

(4)   f.cinax. 

% 

77.    Ri'fiuiredf, 

1 

-^y 

/.- 

sJia^- 

-ej 

-• 

Lot 

a 

tlien 

xz=az, 
D,x  =  o, 

68  DIFFERENTIAL  CALCULUS.  [AuT.  78. 


/.         ' 


V(l-^-)  a 

Examples. 
Find 

1  1  a; 

78.    Required/^         ^ 


Let  z  =  x  +  VC-^'  +  «^)  ; 

z^  —  2  z;.r  +  .j"  =  or  +  a^ 
2zx  =  z-  —  a^, 

^2  _  „i 


2z    ' 
^(a:^  -f  <r)  —  z  —  x  =  z 


'Iz  2z 


D.x  = 


z-  +  g." 

22^ 


_  /•     2z     2"  +  rf- _  ^  1 


•^•^i^2  ^-7;?^  =-^'i  =  '"g^;  =  log(.c  +  V.»^  +  a'O . 


Find  /; 


Example. 

1 


V(-^-'0 


^7is.    log(.r  -}-  Var*  —  a*) . 


Chap.  V.]  INTE(;RAT10N.  69 

79.  When  the  expression  to  be  integrated  can  he  factored,  tlie 
required  integral  can  often  be  obtained  b}-  the  use  of  a  formula 

deduced  from  D^{hv)=  n />, v  +  vD^ v , 

which  gives  uv  =J\ uD^v  +f^vD^xi 

or  f^uD^v  =  nv—f^vD^u.  [1] 

This  method  is  called  integrating  by  parts. 

(a)    For  example,  required  /^log.r. 

log  a;  can  be  regarded  as  the  product  of  logo;  by  1. 

CaU  log. I'  =  u  and  1  =  B^  f, 

then  I)^u  =  -, 

X 

v  =  x\ 
and  we  have 

y;iog.c  =f  1  log.r  =J\  uD^v  =  ny  — /,  t'-^i" 
=  d;log.r  —JV-  =  x\ogx  —  .r. 

X 
PlXAMPLK. 

Find /,a*  log  a;. 

Suggestion:  Let     loga;=  »  and  .r=  D^v. 

Ans.    ^a-(logx-l) 

80.  Required  f^9\v?x. 

Let  u  =  sin.i-  and  />^/*=slnx, 

ihcn  />^»  =  cosx', 

v=  —  coax, 
/,sin*x  =  —  sin.rci)s.>-  +./;cos'x  ; 


70  DIFFERENTIAL  CALCULUS.  [Art.  81. 

but  cos^a;  =  1  —  sin^ic, 

so  XcCOS^x=f^l—/^shr'x  =  x—f^6m'x 

and  y^sin^a;  =  x  —  sinx'coso;  — y^siu^a;. 

2Xsin^a;  =  x  —  sin  a;  cos  a;. 
Xsin^a;  =  J  (a;  —  sin  a;  cos  a;). 

Examples. 

(1)  Find/^cos^a;.  Ans.   -(.'c  +  sin.Tco.sx). 

(2)  /,sinxcosx.  Atis.   ^-. 

81.    Veiy  often  both  methods  described  above  are  required  in 
the  same  integration. 
(a)    Required f,s\xr'^x. 
Let  sin'"';c  =  y,  * 

then  a;=sin?/; 

DyX  —  cos?/, 

Xsin-'.r  =./;?/  =,/;?/ cosy. 

Let  u  =  y  and  D^ v  =  cosy ; 

then  Z)^?A  =  1, 

v  =  siny, 
and 

/yycosy=ysin7/— /,sin?/=?/siny+cosy=a:sin-'a;+ V(l— ar'). 

An}'  inverse  or  anti-function  can  be  integrated  by  this  method 
if  the  direct  function  is  integrable. 

{h)    Thus,  fJ-'x=j^y=f^yDJy  =  yfy-fJy 

where  y=/-»x. 


Chap.   V.J  INTEGRATION.  Tl 


Examples. 

(1)  FindXcos~*a;.  Ans.    a;cos-*a;  — ^(1  —  x*). 

(2)  /,tan-*x.  Ahs.    a-tair'.i;  —    l()ii(l  4- jr). 
( o )  /, vers" ^x.                   Ans.    (x  —  l)  vers" ^x -{- ^{■2x —jr). 

82.    Sometimes  an  algebraic  transformation,  either  alone  or  in 
combination  with  the  preceding  methods,  is  useful. 

(a)    Required/. 


x-^-a^ 


ar—  a-      2  a  \x  —  a      x  +  a  J 
and,  by  Art.  75  (Ex.), 


^C^. 


1  +  X        _  1       __j_ 


V(i-a-=^)     V(i--^")     V(i--'^) 


V(i--'-) 

can  be  readily  obtained  by  substiditing  2/  =  (1  —  x*), 


and  is  —  V(l  — -i")  ". 

hence  /,  J(jzf^)  =  ^i""'-^'  -  V(l  -  ^) • 

(c)    Required /yj(^ii'-x'). 


72  Dll^FKKENTLVL   CALCULUS.  [Art.  83. 

and         f,^{a'  -x^)  =  «y.         .^  -J\  -~^ 


whence        /,  V («'  -  ^")  =  ^''  «i»~ '  -  -/«     ,  f     .„  ,  by  Art.  77 ; 

o.         V(^'  —  ^'") 

but  /.VCa''  -  ar^)  =  x^{a'  -  or)  +./;    ^^  f     ,,  , 

6y  integration  by  imrts^  if  we  let 

ti  =  ^ (a^  —  x-^)  and  D^v=l. 
Adding  our  two  equations,  we  have 

2y;V((r  -  oy")  =  x-^{a?  -  s?)  +  n^siu-i  ^ ; 

and  .-./xVC"^  -  •*■')  =  -^Ma^-o?  +  a^sin-^^Y 

Examples. 

Find 

(1)  /.V(.^-'  +  «'). 

(2)  /,V(x-2-a2). 
1 


-d«.s. 


-  [x-^J^x^  —  a-)  —  «'■  log(x-  +  Vor  —  a^)  J. 


Applications. 

83,    To  find  the  area  of  a  segment  of  a  circle. 
Let  the  equation  of  the  circle  be 

x^  +  ?r  =  "^» 

and  lot  the  required  segment  be  cut  off  by  the  double  ordinates 
through  (a;o,2/o)  and  (.<",?/) .     Then  the  required  area 

A=2f^y-i-C. 


Cii.vp.  v.] 


INTEGKATION. 


73 


From  the  equation  of  the  eiivle, 

y  =  V(*r -.'-"), 
hence  A  =  2/,  V  (""  -  -t"  )  +  C] 

ami  therefore,  by  Art.  82  (*•) , 

A  =  x-^ia-  -  .1")  +  tr  sin-'  '^  +  C. 

As  the  area  is  measured  from  thc^  ordinate  »/„  to  tlie  onlinatt' y, 
^1  =  0  when  .r  =  Jo  ; 
therefore  0  =  .r„^(<r  —  .r,r)  +  a-sin"'  ^'  -fC, 

C=  —x,t^{(r  —  av)  —  <rs\n    '  — » 
and  we  have 

X  Xn 

A  =  x^{fr  —  .T-)  +  a-sin-' -  —  x„y/{(r  —  x,-)  —  rrsin"'- 
If  j;„=  0,  and  the  segment  bey i us  icith  the  axis  o/Y, 

X 

If,  at  the  same  time,  x=  a,  the  sefpnent  heroines  a  semicircle,  and 


The  area  (jf  the  whole  circle  is  ra'''. 


(t        2 


74  differ?:ntial  calculus.  [Art.  84. 

Examples. 

(1)  Slunv  that,  in  the  ease  of  an  ellipse, 

a-       b- 
the  area  of  a  segment  beginning  with  an}'  ordinate  ?/«  is 

.1=   '    .<-V(a-  — .^")  +  (rsin-'-  — a-oV((r— .r„-)  — u^sin-i'^   . 
"  [_  a  a  ] 

That  if  the  segment  begins  witli  the  minor  axis, 

.4  =  -  r  a;  V(«'  -  •'^■')  +  ^''«i"~ '  -1  • 
Tliat  the  area  of  the  whole  ellipse  is  -ah. 

(2)  The  area  of  a  segment  of  the  hyperbola 

«-      6- 
18  ^  =  -[xV(iv2-a2)-anog(a;+V^:r^«) 

—  x^iyj{x^^—  a-)  +  a-  log  (xq + -s/x^  —  a^)  ] . 
If  x^  =  a,  and  the  segment  begins  at  the  vertex, 

^  =  -  [x  V(a^  —  a-)  —  a=log(a;  +  Va.-^  —  a^)  +  aHoga]. 

84.    Tii  find  the  length  of  any  arc  of  a  circle,  the  coordinates 
of  its  extremities  being  (.ro,?/o)  and  (x,y) . 

By  Art.  52,  .s  = /;V[1  + (A?/)']. 

From  the  (■(iiialion  of  the  circle, 

0^  +  ^=0:', 


I'liAP.  v.]  LNTEGRATION.  76 

we  have  2x+  2 // f)^ .'/  =  0 , 


.+(/).,r=-^-i+i-=« 


=/i-=«/x n r  =  «si"     -+C.     (Alt.  /<.) 


When  a:  =  a-Q,  s  =  0  ; 


hence  0  =  «  sin~'  -  +  C, 


C  =  —  ((Sin   '  -, 


" ' ciii~  ' . 


If  Xi,  =  0,  and  the  arc  is  measured  from  the  highest  point  of  the 

X 

circle,  s  =  rtsin   *-• 

If  the  arc  is  a  quadrant,      x  =  a, 

s=asin    '(1)  =  —  , 

and  the  whole  ciicunifcn'iice  =  '2-a. 

H").     To  find  the  length  of  an  arc  of  the  jxirahola  y*=  2nix. 
AVe  have  '^i/D^j/  =  2  m  ; 


Ti  III 

D,y  =  — ; 


76  DIFFERENTIAL   CALCULUS.  [Art.  85. 

/>  .r  =:-L  =IL^  by  Art.  73  ; 

D^y      m 

s  =  —fy  ^rn'+  y-  ^:^\jJ  V"^'  +  /  +  »'i'log(?/  +  V//r+y-)]  +  ^i 

by  Art.  82,  Ex.  1. 
If  the  arc  is  measured  from  the  vertex, 

s  =  0  when  ?/  =  0  ; 

0=-i-(m-logm)  +  C, 
2  ?u 

C=  —  -m\og7n, 
and         ,^lpV('«'  +  /)      „|„,y  +  V(m'  +  /)]. 

2  [_  7«,  ^  O  ^j,  J 

Example. 

Find  the  length  of  the  arc  of  the  curve  .r^=  27,?/- inchidcd  be- 
tween the  origin  and  the  point  whose  abscissa  is  15. 

Ans.    19. 


P^^  ^^^^^^  y^^^  'i^-e-X  •^  .<t^«x<*^-«-^  ^iTt-'r,'-^^^ 


^^l^'  i''i^2y€^'\.^^Va,6^ 


Of 


nAY 


J  re. 


AT 


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